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University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

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Early  American  Mathematics  Books 


■'   -J.  T  »HS_*      ■  .:    ,?'tfc^» 


INTRODUCTIO]^ 


TO    THE 


NATIONAL  ARITHMETIC, 


ON    THE 


INDUCTIVE    SYSTEM, 


COMBIXING    THE 


ANALYTIC  AND  SYNTHETIC  METHODS; 


IN   WHICH  THE  PRINCIPLES   OF   THE   SCIENCE   ARE  FULLY  EXPLAINER 

AND  ILLUSTRATED. 


DESIGNED  rOR  COMMON  SCHOOLS  AND  ACADEMIES. 


By  benjamin  GREENLEAF,  A.M., 

AUTHOR   OF    THE     "NATIONAL   ARITHMETIC,"     "ALGEBRA,"    "  GEOMETKT,"     ETC. 


NEW  ELECTROTYPE  EDITION, 

WITH     ADDITIONS    AND     IMPROVEMENTS. 


BOSTON: 
PUBLISHED    BY    ROBERT    S.    DAVIS    &    CO. 

NEW  YOKK:    D.    APPLETON    &    CO.,    AND    WILLIAM    WOOD    &    CO. 

PHILADELPHIA:     J.    B.    LIPPIXCOTT   AND    C03IPANY. 

CINCINNATI  :     GEO.    S.     BLANCH.\RD. 

ST.  LOUIS:    KEITH  &  WOODS. 

1865. 


I^'  This  Work  is  an  anthoriz'ed  Text-Booh  for  the  Public 
Schools  of  the  City  and  County  of  Philadelphia.  Also  for 
the  Public  Schools  of  New  York  City. 


<  *»»  t 


GREENLEAr'S  SEEIES  OF  MATHEMATICS. 

1.  NEW  PRIMARY  ARITHMETIC  ;  Or,  JIENTAL  ARITHMETIC,  on  the  Induc- 
tive Plan  i  with  Easy  Exercises  for  the  Slate.     Designed  for  Primary  Schools.     84  pp. 

2.  NEW  INTELLECTUAL  ARITHMETIC,  on  the  Inductive  Plan  ;  being  an  Ad- 
vanced Intellectual  Course,  for  Common  Schools  and  Academies.     180  pp. 

3.  COMMON  SCHOOL  ARITHMETIC  ;  Or,  INTRODUCTION  TO  THE  NATIONAL 
ARITHMETIC.    A  Complete  Treatise.     Improved  electrotype  edition,    324  pp. 

4.  THE  NATIONAL  ARITHMETIC,  being  a  complete  Course  of  Higher  Arithmetic, 
for  advanced  Scholars  in  Common  Schools,  High  Schools,  and  Academies.  New  electro- 
type edition,  with  additions  and  improvements     444  pp. 

5.  NEW  ELEMENTARY  ALGEBRA  ;  in  which  the  First  Principles  of  Analysis  are 
progressively  developed  and  simplified.    For  Common  Schools  and  Academies.    324  pp. 

6.  NEW  HIGHER  ALGEBRA  ;  an  advanced  Analytical  Course,  for  High  Schools, 
Academies,  and  Colleges.    394  pp.     [Just  Published.] 

7.  ELEMENTS  OF  GEOMETRY,  with  Practical  Applications  to  Mensuration.  320  pp. 

8.  ELEMENTS  OF  TRIGONOMETRY,  with  Practical  Applications,  and  Tables. 

9.  ELEMENTS  OF  GEOMETRY  AND  TRIGONOMETRY  ;  or  the  last  two  named 
works  in  one  volume.    490  pp. 

10.  TREATISE  ON  SURVEYING  AND  NAVIGATION  ;  with  Practical  Applications 
and  Tables.     [In  preparation.] 

O"  KEYS  to  the  Arithmetics,  Algebras,  Qeometbt  and  Teigosjometrt.  For  Teach- 
ers only.    6  volumes. 


i  ^»^  > 


Entered  according  to  Act  of  Congress,  in  the  year  1842,  by 

BENJAMIN     G  R  K  E  N  L  E  A  F , 

in  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


Entered  according  to  Act  of  Congress,  in  the  year  1848,  by 

B  E  N  .1  A  M  I  N     G  K  E  E  N  L  E  A  F  , 

in  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


Entered  according  to  Act  of  Congress,  in  the  year  1856,  by 

B  E  N  .1  A  M  I  N     G  R  E  E  N  L  E  A  F  , 

In  tho  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


PREFACE. 


The  present,  edition  of  this  work  has  been  thoroughly  revised 
and  re-written,  and  also  improved  by  the  addition  of  much  valu- 
able new  matei'ial,  rendering  it  a  sufficiently  complete  practical 
ti'eatise  for  the  majority  of  learners. 

The  arrangement  is  strictly  progressive  ;  the  aim  having  been 
to  introduce  subjects  in  an  order  most  in  accordance  with  the 
laws  governing  the  proper  development  of  mind.  The  rules  have 
generally  been  deduced  from  the  analysis  of  one  or  more  ques- 
tions, so  that  the  reasons  for  the  methods  of  solution  adopted  are 
rendered  intelligible  to  the  pupil ;  no  knowledge  of  a  principle  be- 
ing required,  that  has  not  been  previously  illustrated  and  explained. 

In  preparation  of  the  rules,  definitions,  and  illustrations,  the 
utmost  care  has  been  taken  to  exjjress  them  in  language  simple, 
precise,  and  accurate. 

The  examples  are  of  a  practical  character,  and  adapted  not 
only  to  fix  in  the  mind  the  principles  which  they  involve,  but 
also  to  interest  the  pupil,  exercise  his  ingenuity,  and  inspii-e  a 
love  for  mathematical  science. 

The  reasons  for  the  operations  are  explained,  and  an  attempt 
is  made  to  secure  to  the  learner  a  knowledge  of  the  philosophy  of 
the  subject,  and  prevent  the  too  prevalent  practice  of  merely  per- 
forming, mechanically,  operations  which  he  does  not  understand. 

Analysis  has  been  made  a  prominent  subject,  and  employed  in 
the  solution  of  questions  under  most  of  the  rules  in  which  it 
could  be  used  with  any  practical  advantage. 

All  the  most  important  methods  of  abridging  operations,  appli- 
cable to  business  transactions,  have  been  given  a  place  in  the 
work,  and  so  introduced  as  not  to  be  regarded  as  mere  blind 
mechanical  expedients,  but  as  rational  labor-saving  processes. 

Old  rules  and  distinctions,  which  modern  improvements  have 
rendered  unnecessary,  and  which,  deservedly,  are  becoming  obso- 
lete, have  been  avoided. 


IV  PREFACE. 

Rules  for  finding  the  Greatest  Common  Divisor  of  Fractions, 
and  lor  finding  the  Least  Common  Mvltiple  of  Fractions  ;  Meth- 
ods of  Equating  Accounts  ;  Division  of 'Duodecimals  ;  Exchange, 
Foreign  and  Inland  ;  and  several  important  Tables,  are  among  the 
new  figatures  of  this  edition,  which  will  be  found,  it  is  beheved, 
very  valuable. 

The  articles  on  Money,  "Weights,  Measures,  Interest,  and  Du- 
ties are  the  results  of  extensive  correspondence  and  much  labori- 
ous research,  and  are  strictly  conformable  to  present  usage,  and 
recent  legislation.  State  and  national. 

The  interpretation  of  Ratio  adopted  in  this  work  is  the  simple 
and  natural  method  of  Chauvenet,  Peirce,  Loomis,  Hackley,  Al- 
sop,  Day,  and  other  prominent  mathematicians  of  this  country, 
and  of  nearly  all  European  authorities,  including  Sir  Isaac  New- 
ton, Laplace,  Legendre,  and  Bessel. 

Questions  have  been  inserted  at  the  bottom  of  each  page,  de- 
signed to  direct  the  attention  of  teachers  and  pupils  to  the  most 
important  principles  of  the  science,  and  fix  them  in  the  mmd.  It 
is  not  intended,  however,  nor  is  it  desirable,  that  the  teacher 
should  servilely  confine  himself  to  these  questions  ;  but  vary  their 
form  and  extend  them  at  pleasure,  and  invariably  require  the 
pupil  thoroughly  to  understand  the  subject. 

The  object  of  studying  mathematics  is  not  only  to  acquire  a 
knowledge  of  the  subject,  but  also  to  secure  mental  discipline,  to 
induce  a  habit  of  close  and  patient  thought,  and  of  persevering 
and  thorough  investigation.  For  the  attainment  of  this  object,  the 
examples  for  the  exercise  of  the  pupil  are  numerous,  and  variously 
diversified,  and  so  constructed  as  necessarily  to  require  careful 
thought  and  reflection  for  the  right  application  of  principles. 

The  author  would  respectfully  suggest  to  teachers,  who  may 
use  this  book,  to  require  their  pupils  to  become  familiar  with  each 
rule  before  they  proceed  to  a  new  one  ;  and,  for  this  purpose,  a 
frequent  review  of  rules  and  principles  will  be  of  service,  and 
will  greatly  facilitate  their  progress. 

BENJAMIN  GREENLEAF. 


NOTICE. 

Two  editions  of  this  work,  and  also  of  the  National  Arith- 
METic,  one  containing  tlio.  answers  to  the  examples,  and  the  other 
without  them,  arc  now  published. 


CONTENTS. 


SIMPLE    NUMBERS. 


PAGE 

Definitions 7 

Notation      •• 7 

Exercises  in  Eoman  Notation    .     .  9 

Numeration 11 

Freiicli  Numeration  Table    ...  11 

Exercises  in  Frencli  Numeration  .  12 

English  Numeration  Table    ...  14 

Exercises  in  En2:lisli  Numeration  .  15 


PAGE 

Addition .16 

Subtraction 25 

Multiplication 83 

Division 44 

57 
61 
63 
65 


Questions  involving  Fractions 
Contractions  in  Multiplication 
Contractions  in  Division  .    . 
^liscellaueous  Examples  .     . 


UNTTED 

United  States  Money 69 

Table 69 

Reduction 70 

Addition 71 

Subtraction 73 


STATES    MONEY. 

]\Iultiplicatiou 74 

Division 75 

Analysis 76 

Bills 79 

Ledjrer  Accounts 81 


COMPOUND 

Eeduction 82 

English  Money 82 

Troy  Weight 84 

Apothecaries'  Weight 80 

Avoirdupois  Weight 87 

Cloth  Measure 89 

Long  or  Linear  Measure   ....  90 

Surveyors'  Measure 93 

Surface  or  Square  Measure  ...  94 

Cubic  or  Solid  Pleasure    ....  96 

Wine  or  Liquid  Measure  ....  98 

Beer  Measure 99 


NUMBERS. 

Dry  Jleasnre 100 

Measure  of  Time 102 

Circular  Measure 105 

Jliscellaneous  Table 106 

Miscellaneous  Exercises   ....  107 

Addition 110 

Subtraction 114 

Miscellaneous  Exercises  ....  119 

Multiplication 121 

Division 125 

Miscellaneous  Exercises  in  Multi- 
plication aiid  Division  .     .     .  129 


PROPERTIES    OF    NUMBERS. 

Properties  of  Numbers  ....  130 
Table  of  Prime  Numbers  ....  131 
Cancellation 133 


Common  Divisor 136 

Greatest  Common  Divisor    .     .     .  136 
Common  Multiple 138 


COMMON    FRACTIONS. 

Common  Fractions 140  i  Addition 148 

Reduction 142  |  Subtraction     .     , J 50 

Common  Denominator      ....  146  j  ilultiplication 155 


vi 


COJJTEKTS. 


Division 


Complex  Fractions  .  .  . 
Greatest  Common  Divisor 
Least  Common  Multiple  . 
Miscellaneous  Exercises  . 
lieduction  of  Fractions  of  Com- 
pound Numbers 170 


160 
165 
167 
167 
169 


Addition  of  Fractions  of  Compound 

Numbers 174 

Subtraction  of  Fractions  of  Com- 
pound Numbers    .    .    .    .    .  175 

Questions  by  Analysis      ....  176 

Miscellaneous  Questions  ....  179 


BECIMAL    FRACTIONS. 


Numeration 182 

Notation 183 

Addition 184 

Subtraction 185 


Multiplication 186 

Division ^     .     .  188 

Reduction 190 

Miscellaneous  Exercises  ....  193 


PERCENTAGE. 


Percentage 194 

Simple  Interest 196 

Miscellaneous  Exercises   ....  204 

Partial  Payments 205 

Problems 210 

Compound  Interest 212 

Discount 216 

RATIO    AND 

Ratio 237 

Proportion 239 

Simple  Proportion 240 


Commission,  Brokerage,  and  Stocks  218 

Banking 221 

.  222 
.  224 
.  225 
.  227 
,  230 


Bank  Discount    .     .    . 

Insurance    

Custom-House  Business 
Assessment  of  Taxes  . 
Equation  of  Payments 


PROPORTION. 

Compound  Proportion  .....  245 

Profit  and  Loss 248 

Partnership,  or  Company  Business  25* 


CURRENCIES. 


Reduction  of  Currencies 

Exchange    

Inland  Bills     .     .  "  .    . 


259 
261 
262 


Foreign  Bills 263 

Exchange  on  England 263 

Exchange  on  France 265 


Addition  and  Subtraction 


DUODECIMALS. 

.    .  266  I  Multiplication  and  Division  . 


INVOLUTION    AND 

Involution *  ....  209 

Evolution 271 

Extraction  of  Square  Root   .    .    .  272 


EVOLUTION. 

Application  of  Square  Root . 
Extraction  of  Cube  Root  . 
Application  of  Cube  Root 


MISCELLANEOUS. 

Arithmetical  Progression ....  287 
Annuiiies  at  Simple  Interest  .  .  292 
Geometrical  Progression  ....  294 
Annuities  at  Compound  Interest  .  298 
Allegation 300 


Allegation  Modlal 300    Miscellaneous  Questions 


Allegation  Alternate    . 
Permutation    .... 
Mensuration  of  Surfaces 
Mensuration  of  Solids  . 
Mensuration  of  Lmnber 


267 


276 
281 
285 


301 
305 
306 
312 
318 
319 


AEITHMETIC. 


DEFINITIONS. 

Article  1.     Quantity  is  anything  that  can  be  measured. 

A  Unit  is  a  single  thing,  or  one. 

A  Number  is  a  unit  or  a  collection  of  units. 

An  Abstract  Number  is  a  number,  whose  units  have  no  reference 
to  any  particular  thing  or  quantity  ;  as  two,  five,  seven. 

A  Concrete  Number  is  a  number,  whose  units  have  reference  to 
some  particular  thing  or  quantity  ;  as  two  books,  five  feet. 

The  Unit  of  a  Number  is  one  of  the  same  kind  as  the  numlier ; 
thus,  the  unit  of  six  is  one,  and  the  unit  of  six  pounds  is  one  pound. 

Arithmetic  is  the  science  of  numbers,  and  the  art  of  computing 
by  them. 

A  Rule  is  a  prescribed  mode  for  performing  an  operation. 

The  Introductory  Processes  of  arithmetic  are  Notation,  Numera- 
tion, Addition,  Subtraction,  Multiplication,  and  Division. 

The  last  four  are  called  the  fundamental  rules,  because  upon 
them  depend  all  other  arithmetical  processes. 


NOTATION, 


2 1    Notation   is   the   art  of  expressing  numbers  by  figures  or 
other  symbols. 

There  are  two  methods  of  notation  in  common  use ;  the  Roman 
and  the  Arabic. 


QuESTiOKS.  —  Art.  1.  What  is  quantity?  A  unit?  A  number?  An 
abstract  number  ?  A  concrete  number  ?  Arithmetic  ?  A  rule  ?  Which 
are  the  introfluctory  processes  ?  Wliat  are  the  last  four  called  ?  —  2.  What  is 
notation  ?     How  many  kinds  of  notation  in  common  use  ?     What  are  they  ? 


8 


NOTATION. 


3.  The  Roman  Notation,  or  that  originated  by  the  ancient 
Konians,  emplojd  in  expressuig  numbers  seven  capital  letters, 
viz. : — 

I,      V,      X,      L,  C,  D,  M. 

one,   five,    ten,    fifty,    one  hundred,    five  hundred,   one  thousand. 

All  the  other  numbers  are  expressed  by  the  use  of  these  let- 
ters, either  in  repetitions  or  combinations. 

1.  By  a  repetition  of  a  letter,  the  value  denoted  by  the  letter 
is  repeated;  as,  XX  represents  twenty;  CCC,  three  hundred. 

2.  By  writing  a  letter  denoting  a  less  value  before  a  letter  de- 
noting a  greater,  the  difference  of  their  values  is  represented  ;  as, 
IV  represents /oMr;  '^Ij,  forty. 

3.  By  writing  a  letter  denoting  a  less  value  after  a  letter  de- 
noting a  greater,  the  sum  is  represented  ;  as,  VI  represents  six  ; 
XV,  fifteen. 

4.  A  dash  ( — )  placed  over  a  letter  makes  the  value  denoted  a 
thousand-fold;  as,  V  represents  ^re  thousand  ;  \Y,four  thousand. 

TABLE. 


I 

one. 

LXXX 

eighty. 

11 

two. 

XC 

ninety. 

III 

three. 

c 

one  hundred. 

IV 

four. 

cc 

two  hundred. 

V 

five. 

CCC 

three  hundred. 

VI 

six. 

cccc 

four  hundred. 

VII 

seven. 

D 

five  hundred. 

VIII 

eight. 

DC 

six  hundred. 

IX 

nine. 

DCC 

seven  hundred. 

X 

ten. 

DCCC 

eight  hundred. 

XX 

twenty. 

DCCCC 

nine  hundred. 

XXX 

thirty. 

M 

one  thousand. 

XL 

forty. 

MD 

fii'tecii  lumdred. 

L 

fifty. 

M]\I 

two  thousand. 

LX 

sixty. 

X 

ten  thousand. 

LXX 

seventy. 

]\I 

one  million. 

3.  Whv  is  the  Roman  notation  so  called  ?  By  what  arc  numbers  expressed 
in  the  Koman  notation  1  "What  effect  has  the'  repetition  of  a  letter  •?  Tlie 
cli'ect  of  writing'  a  letter  expressinp:  a  less  value  before  a  letter  dcnotiuLr  a 
{jrcatcr  ?  Of  writing;  a  letter  after  anotiier  denoting  a  <rreater  value  ?  IIuw 
many  fold  is  the  value  denoted  by  a  letter  made  by  placing  a  dash  over  iti 
licpeat  the  table. 


NOTATION.  9 

The  Roman  notation  is  now  but  little  used,  except  in  number- 
ing sections,  chapters,  and  other  divisions  of  books. 

Exercises  in  Roman  Notation. 

"Write  the  following  numbers  in  letters  :  — 

1.  Ninety-six.  Ans.  XCVI. 

2.  Eighty-seven. 

3.  One  hundred  and  ten. 

4.  One  hundred  and  sixty-nine. 

5.  Two  hundred  and  seventy-five. 

6.  Five  hundred  and  forty-two. 

7.  One  thousand  three  hundred  and  nineteen. 

8.  One  thousand  eight  hundred  and  fifty-eight. 

4,  The  Arabic  Notation,  or  that  made  known  through  the  Arabs, 
employs  in  expressing  numbers  ten  characters  or  figures,  viz. :  — 

1,       2,        3,         4,        5,       6,        7,  8,         9,  0. 

one,    two,    three,    four,    five,    six,    seven,    eight,    nine,    cipher. 

The  first  nine  are  called  digits,  from  digitus,  the  Latin  signify- 
ing a  finger,  because  of  the  use  formerly  made  of  the  fingers  in 
reckoning.  The  cipher  is  called  naught,  or  zero,  from  its  express- 
ing the  absence  of  a  number,  or  nothing,  when  standing  alone. 

5t  The  particular  position  a  figure  occupies  with  regard  to 
other  figures  is  called  its  place  ;  as  in  32  (thirty-two),  counting 
from  the  right,  the  2  occupies  the  first  place,  and  the  3  the  second 
place. 

The  digits  have  been  denominated  significant  figures,  because 
each  of  itself  always  represents  so  many  units,  or  ones,  as  its 
name  indicates.  But  the  size  or  value  of  the  units  represented 
by  a  figure  differs  according  to  the  place  occupied  by  it. 

Thus,  in  366  (three  hundred  and  sixty-six),  each  of  the  fig- 
ures, without  regard  to  its  place,  represents  units,  or  ones  ;  but 
the  6  occupying  the  first  place  represents  6  single  units ;  the  6 

.3.  What  use  is  now  made  ofRonian  notation  ?  — 4.  How  many  characters 
are  employed  in  the  Arabic  notation  1  What  are  the  first  nine  called,  and 
why?  The  cipher?  What  does  it  represent  when  standing  alone?  — 
5.  Wliat  is  meant  by  the  place  of  a  figure  ?  What  have  the  digits,  been 
denominated  ?  Why  ?  How  does  the  size  or  value  of  units  represented  by- 
figures  differ  ? 


10  NOTATION. 

occupying  the  second  place  represents  G  tens,  or  G  units  each 
ten  times  the  size  or  v.ikie  of  a  unit  of  the  first  place ;  and  the 
3  oci'ui)ying  the  third  place  represents  3  hundreds,  or  3  units 
each  one  hundred  times  the  size  or  value  of  a  unit  of  the  first 
place. 

6.  The  cipher,  when  connected  with  other  figures,  occupies  a 
place  that  otherwise  would  be  vacant;  as  in  10  (ten),  where  it 
occupies  the  vacant  place  of  units ;  and  in  304  (three  hundred 
and  four),  where  it  occupies  the  vacant  place  of  tens. 

7.  The  Simple  Value  of  a  unit  is  the  value  expressed  by  a  fig- 
ure standing  alone  ;  or,  in  a  collection,  when  standing  in  the 
right-hand  place. 

Thus  6  alone,  or  in  26,  expresses  a  simple  value  of  six-  single 
units,  or  ones. 

The  Local  Value  of  a  unit  is  the  value  expressed  by  a  figure 
when  it  is  used  in  combination  with  another  figure  or  firjures,  and 
depends  upon  the  place  the  figure  occupies. 

Tlie  local  values  expressed  by  figures  will  be  made  plain  by 
the  following 


TABLE. 


'2  CO 

C  -73 

^-  a 

S  '^ 

2  3 

-^     o 

H  r-"    w 

, 

c     •   o   « 

!-i 

•^t    2    1 

c 

3 

2  -2 

o  '3 

ji^HS 

HP 

9 

9  8 

9  8  7 

9 

8  7  6 

9  8 

7 

G  5 

9  8  7 

6 

5  4 

9  8  7  6 

5 

4  3 

The  figures  in  this  table  are  read  thus  :  — 


Xine. 

Ninety-eight. 

Nine  hundred  eighty-seven. 

Nine  thousand  eight  hundred  seventy-six. 

Nincty-ciglit  thousand  seven  hundred  sixty-five. 

5  Nine  Imndred  eighty-seven  thousand  six  liundi'ed 
(       fifty-four. 

(  Nine  millions  eitrht  himdred  seventy-six  thousand 
\       five  hundred  fbrty-thn^e. 

6*  What  docs  a  ciplior  occupy  wlicn  written  in  connection  with  otiicr  fig- 
ures? —  7.  Wlint  is  the  simple  viiluc  of  a  miit,?  Tin;  local  value  of  a  unit  i 
The  dosiirn  of  the  tahle  1 


NUMERATION. 


11 


In  the  table,  any  figure  in  the  right-hand  or  units'  place  ex- 
pi'csses  the  local  value  of  so  many  units ;  but  the  same  in  the 
second  place  expresses  the  local  vahie  of  so  many  tens,  each  of 
the  value  of  ten  ones ;  in  the  third  place,  the  local  value  of  so 
many  hundreds,  each  of  the  value  of  ten  tens  ;  in  the  fourth  place, 
the  local  value  of  so  many  thousands,  each  of  the  value  of  ten 
hundreds  ;  and,  in  general, 

TJie  value  expressed  In/  any  figure  is  ahoays  made  tenfold  by 
each  'removal  of  it  one  place  to  tlie  left  hand. 


NUMERATION. 

8t  NlimeratiuU  is  the  art  of  reading  numbers  vv^hen  expressed 
by  figures. 

9i  There  are  two  methods  of  numeration  in  common  use : 
the  French  and  the  English. 

10.  The  Freacll  Method  is  that  in  general  use  on  the  continent 
of  Europe  and  in  the  United  States.  It  separates  figures  into 
groups,  called  periods,  of  three  places  each,  and  gives  a  distinct 
«ame  to  each  period. 

FRENCH  NUMERATION  TABLE. 


o 


tT2    m 

Sext 
illion 

QuIr 
itillio 

o    ^     , 

r^       ^       ^ 

oj ««-  .2 

5  "  X 

SHcc 


o  "5 


^  Ol 


^     ^     ^ 

^     O     —^ 


c 
.2 

1:2 

P     QJ     P 

WHO* 


r^ 

C 

O 

H 

O 

«*- 

■-^ 

o 

"C 

w 

H 

. 

<D  5j- 

m 

o 

O 

fl 

r- 

K 

12  7,     8  9  4,    2  3  7,     8  G  7, 


s 
.2 

r^  2 
c  :::3 
CO  rn 

^3  I— I 


KHcq 
1  2  3, 


o 


o 


W 
6 


o 


8, 


m 

0 

2  fl 
^  I    . 

k3 


i:   IK   S 
£  a  o 

4  7  8, 


"I  S  -2 
G  3  8. 


Period  of 
Sextil- 
lions. 


Period  of 

Quintil- 

Uons. 


Period  of 

Qiuidiil- 

lious. 


Period  of 
Trillions. 


Period  of 
Billions. 


Period  of      Period  of         Period  of 
llilUous.     Thousands.         Units. 


7.  "What  value  is  expressed  by  a  fif^ure  standing  in  the  right-hand  or  units' 
place  ?  In  the  second  place  1  In  the  third  "(  How  do  tigures  increase  fiom 
the  right  towards  the  left?  —  8.  What  is  numeration  ?  —  9^  What  are  the  two 
methods  of  numeration  in  common  use  ?  —  10.  Where  is  th.e  French  method 
more  generally  used  ?  Kepeat  the  French  Numeration  Table.  Name  the 
diiFerent  periods  in  the  table. 


12 


XUJrEEATION. 


The  value  of  the  number  represented  in  the  table  is,  One 
hundred  twenty-seven  scxtillions,  eiglit  hundred  ninety-four  qinn- 
tillions,  two  hundred  thirty-seven  quadrillions,  eight  hundred 
sixty-seven  trillions,  one  liundred  twenty-three  billions,  six  liun- 
drcd  seventy-eight  millions,  four  hundi'ed  seventy-eight  thousand, 
six  hundred  thirty-eight. 

The  periods  above  SextilHon?,  in  their  order,  are,  Septillions, 
Octillions,  NonilUons,  Decillions,  Undecillions,  Duodecillions,  Tre- 
decillions,  Quatuordecillions,  Quindecillions,  Sexdecillions,  Septen- 
deciUious,  Octodecillions,  .Novemdecillions,  Vigintilhons,  &c. 

1 1 ,  The  successive  places  occupied  by  figures  are  often  called 
Orders.  A  figure  in  the  right-hand  or  units'  place  is  called  a  figure 
of  the  Jirst  order,  or  of  the  order  of  U7iifs  ;  a  figure  in  the  second 
place  is  a  figure  of  the  second  order,  or  of  the  order  of  tens  ;  hi 
the  third  place,  of  the  order  of  hundreds,  and  so  on. 

Thus,  in  1847,  the  7  is  of  the  order  of  units,  4  of  the  order  of 
tens,  8  of  the  order  of  hundreds,  and  1  of  the  order  of  thousands, 
so  that  we  read  the  w^hole,  one  thousand  eight  hundred  and  forty- 
seven. 

12.  To  numerate  and  read  figures  according  to  the 
French  method. 

RuLB.  —  Begin  at  the  right,  and  point  off  the  figures  into  periods  of 
THREE  places  each. 

Tlien,  commencing  at  the  left,  read  the  figures  of  each  period,  giving 
the  name  of  each  period  excepting  that  of  units. 

Exercises  m  French  Numeration. 


Read  or  write  in  words  the  numbers  represented  by  the  follow- 
ing figures :  — 


1. 

152 

5. 

2254 

9. 

84093 

13. 

610711 

2. 

276 

6. 

4384 

10. 

98612 

14. 

3031671 

3. 

998 

7. 

7932 

11. 

592614 

15. 

4869021 

4. 

1057 

8. 

42198 

12. 

400619 

16. 

637313789 

10.  Wlmt  is  tlic  value  of  the  number  represented  in  the  talile  expressed  in 
words  ?  Wl)at  are  the  names  of  tlic  periods  above  sextillions  ?  —  11.  What 
are  the  successive  places  of  the  futures  in  the  table  called?  Of  what  order 
is  the  first  or  ri<rht-hand  figure  ?  The  second  ?  The  third  ^  &c.  —  12.  The 
rule  for  numeratimj  and  reading  numbers  according  to  the  French  method  ? 


■ 

SIUilERAT 

ION. 

13 

17. 

S9461928 

24. 

8761700137706717 

18. 

427143271 

25. 

242173562357421 

19. 

6301706716 

26. 

870037637471078635 

20. 

143776700333 

27. 

8216243812706381 

21. 

20463162486135 

28.   . 

2403172914376931 

22. 

63821024711802 

29. 

3761706137706167138 

23. 

44770630147671 

30. 

610167637896430607761607 

13.    To  write   numbers   by   figures    according   to    the 
French  method. 

Rule.  —  Begin  at  the  left,  and  write  in  each  successive  order  the  fig- 
ure belonging  to  it. 

If  any  order  ivould  otherwise  he  vacant,  fill  the  place  hij  a  cipher. 

Exercises  in  French  Notation  and  Numeration. 

Represent  by  figures,  and  read,  the  following  numbers  :  — 

'1.  Foi'ty-seven. 

2.  Three  hundred  fifty-nine. 

3.  Six  thousand  five  hundred  seventy-five. 

4.  Nine  hundred  and  eight. 

5.  Nineteen  thousand.  . 

6.  Fifteen  hundred  and  four. 

7.  Twenty-seven  millions  five  hundred. 

8.  Ninety-nine  thousand  ninety-nine. 

9.  Forty-two  millions  two  thousand  and  five. 

10.  Four  hundred  eight  thousand  ninety-six. 

11.  Five  thousand  four  hundred  and  two. 

12.  Nine  hundred  seven  millions  eight  hundred  five  thousand 
and  seventy-four. 

13.  Three   hundred  forty-seven  thousand   nine   hundi-ed    and 
fifteen. 

14.  Eighty-nine  thousand  forty -seven. 

15.  Fifty-one  thousand  eighty-one. 

16.  Seven  thousand  three  hundred  ninety-five. 

17.  Fifty -seven   billions    fifly-niue   millions    ninety -nine   thou- 
sand and  forty-seven. 

13.   The  rule  for  writing  numbers  according  to  the  Frencli  method  ?     At 
wliich  hand  do  you  he;,nn  to  numerate  ?     Wliere  do  you  begin  to  read  ?    At 
which  hand  do  you  betrin  to  write  numbers  '>     Why  ?" 
2 


14  NUMERATION. 

14i  The  Elijrlish  Illclliod  of  numeration  separates  figures  into 
periods  of  six  figures  each.  The  first  or  right-hand  period  is  re- 
garded as  units  and  thousands  of  units  ;  the  second,  as  millions 
and  thousands  of  millions  ;  and  so  on,  six  places  being  assigned 
to  each  period  designated  by  a  distinct  name. 


ENGLISH  NUMERATION   TABLE. 


§s  s»  i§ 

s|  jij  1.1 

„oSg  wjogm  „ocg  S 


c  " 


-.2  'Tmcg  tJjo.Sg  gm 


rtc't-?"  rtcr;==3.  «c!L5r2a5  2a 

p^tHHo  P??Wqqa  pK'^^SS  Ht« 

C3v.,:3  cst^Po  ost^-p'.o  ^3 

C2^^?  e52°^S  eS2°^3  ^2. 

oc^h-cjc^hS       Cv-5o«t-»;       cc«'-(uv<2       o^So 
.-jOS^og        ^Ort^Oc        ^o«^og        ^o^^ 

Scogsrp       §r2-?3       Scgn?.:3       s?2sp'a 
KHHKHEh      GHHSHM      KHHWH^      KHHSHP 

13789  0,      711716,     371712,     456711. 

V ^ '       V ^ ;       » ^ ;       V ^ , 

Period  of  Tril-  Period  of  Bil-  Period  of  Mil-  Period  of 

lions.  lions.  lions.  Units. 

The  value  of  the  figures  in  the  table,  according  to  the  English 
method,  is,  One  hundred  thirty-seven  thousand  eight  hundred 
ninety  trillions,  seven  hundred  eleven  thousand  seven  hundred 
sixteen  billions,  three  hundred  seventy-one  thousand  seven  hun- 
dred twelve  millions,  four  hundi'ed  fifty-six  thousand  seven  hun- 
dred eleven. 

Note.  —  Altliouq;li  tlicre  is  the  same  numhcr  of  fijruros  in  the  Enjilish  and 
in  tlie  French  tal)le,  yet  it  will  be  observed  tliat  in  tiic  French  table  we  have 
the  names  of  tlii-ee  periods  other  than  in  the  Etiijlish.  It  will  also  be  observed 
that  the  variation  commences  after  the  ninth  place,  or  the  plafc  of  hundreds 
of  nulUons.     If,  therefore,  we  would  know  the  value  of  numbers  higher  than 


1  t.  ITo\v  many  fijures  in  each  porind  by  the  English  metlind  of  numeration  ? 
Wliat  ordi'rs  are  found  in  tlic  ICn^^lish  method  that  arc  not  in  tiie  French  ? 
Give  the  names  of  the  jieriods  in  the  table,  betrinniiii;  with  unit';.  I^epeat 
the  table.  What  is  the  value  of  the  numbers  in  the  table  ?  How  do  the 
fi;_'tirps  in  the  Enjrlish  and  French  table  compare  as  to  nninbcrs  ?  How  as  to 
periods?  Why  is  this  difTi'rence ?  Ha?  a  million  the  same  value  reckoned 
by  the  French  table  as  when  reckoned  by  the  English  ? 


NUMERATION.  1 


n 


hundreds  of  millions,  when  we  see  them  written  in  words,  or  hear  them  read, 
we  need  to  know  whether  they  are  expressed  according  to  the  French  or  the 
English  method  of  numeration. 

The  Englisli  method  of  nmncration  is  that  generally  used  in  Great  Britain, 
and  in  the  British  Provinces. 

15.    To  numerate  and  read  figures  according  to  the 
Englisli  method. 

Rule. — Begin  at  the  right,  and  point  off  the  figures  into  periods  of 
SI'S,  places  each.  Then,  commencing  at  the  left  hand,  read  the  figures  of 
each  jicriod,  giving  the  name  of  each  pei'iod,  excepting  that  of  units. 

Exercises  in  English  Numeration. 

Read  orally,  or  write  in  words,  the  numbers  represented  by 
tlie  following  figures  :  — 


1. 

125 

5. 

2730G387903 

2. 

1063 

6. 

531470983712 

3.  ' 

25842 

7. 

4230578032705038 

4. 

904357 

8. 

71675637880737076708G3897O6473 

16.    To   write   numbers   in   figures   according   to   the 
English  method. 

Rule.  —  Begin  at  the  left,  and  write  in  each  successive  order  the 
figure  belonging  to  it. 

If  any  order  would  otherwise  he  vacant,  fill  the  place  hy  a  cipher. 

Exercises  in  English  Notation  and  Numeration. 

"Write  in  figures,  and  read,  tbe  folloAving  numbers  :  — 

1.  Three  hundred  twenty -five  thousand  four  hundred  and 
twelve. 

2.  Two  hundred  fourteen  thousand,  one  hundred  sixty-five 
millions,  seventy-eight  thousand  and  fifty-six. 

3.  Forty-two  billions,  six  hundred  seventeen  thousand  thirty- 
one  millions,  forty-one  thousand  three  hundred  forty-two, 

4.  Two  thousand  eight  billions,  nine  thousand  eighty-two 
millions,  seven  hundred  one  thousand,  nine  hundred  and  eight. 

14.  Ha'!  the  billion  the  s.ime  value  as  that  hv  the  French  table?  Why 
not  ?  By  which  table  has  it  the  creatcr  value  "?  Where  is  the  English  method 
of  niimeratint;  in  use?  —  15.  Wliat  is  the  rule  for  numerating  and  reading 
numbers  by  the  English  method  ?  —  16.  The  rule  for  writing  numbers  accord- 
ing to  the  English  method  ? 


16 


ADDITION. 


ADDITION. 

17t  When  it  is  required  to  find  a  single  number  to  express 
the  sum  of  the  units  contained  in  several  smaller  numbers  of  the 
same  kind,  the  process  is  called  AddiUoil. 

Ex.  1.  James  has  3  pears,  and  his  younger  brother  has  4  ; 
how  many  have  both  ? 

Illustration.  —  The  3  pears  and  4  pears  must  be  added ;  thus, 
3  peaj'3  added  to  4  pears  make  7  pears.     Therefore  both  have  7  pears. 


ADDITION    TABLE. 


2  and 

0  are 

2 

3  and 

0  are 

3 

4  and 

0  are 

4 

5  and 

0  are  5 

2  " 

1  " 

3 

3  " 

1  " 

4 

4 

a 

1  " 

6 

5  " 

1  "   6 

2  " 

2  " 

4 

3  " 

2  " 

5 

4 

u 

2  " 

6 

5  " 

2  "  7 

2  " 

3  " 

5 

3  " 

3  " 

6 

4 

a 

3  " 

7 

5  " 

3  "  8 

2  " 

4  " 

6 

3  " 

4  " 

7 

4 

a 

4  " 

8 

5  " 

4  "  9 

2  " 

6  " 

7 

•  3  " 

5  " 

8 

4 

u 

5  " 

9 

5  " 

5  "  10 

2  " 

6  " 

8 

3  " 

6  " 

9 

4 

£t 

6  " 

10 

5  " 

6  "  11 

2  « 

7  " 

9 

3  " 

7  " 

10 

4 

It 

7  " 

11 

5  " 

7  "  la 

2  " 

8  " 

10 

3  " 

8  " 

11 

4 

li 

8  " 

12 

5  " 

8  "  13 

2  " 

9  " 

11 

3  " 

9  " 

12 

4 

u 

9  " 

13 

5  " 

9  "  14 

2  " 

10  " 

12 

3  " 

10  " 

13 

4 

u 

10  " 

14 

5  " 

10  "  15 

2  " 

11  " 

13 

3  " 

11  " 

14 

4 

(( 

11  " 

15 

5  " 

11  "  16 

2  " 

12  » 

14 

3  " 

12  " 

15 

4 

u 

12  " 

16 

5  " 

12  "  17 

6  and 

0  are 

6 

7  and 

0  are 

7 

8 

and 

0  are 

8 

9  and 

0  are  9 

G  " 

1  " 

7 

7  " 

1  " 

8 

8 

u 

1  " 

9 

9  " 

1  "  10 

6  " 

2  ". 

8 

7  " 

2  " 

9 

8 

u 

O  " 

10 

9  " 

2  "  11 

6  '' 

3  " 

9 

7  » 

3  " 

10 

8 

(C 

3  " 

11 

9  " 

3  "  12 

6  " 

4  " 

10 

7  " 

4  " 

11 

8 

u 

4  " 

12 

9  " 

4  "  13 

0  " 

5  " 

11 

7  " 

6  " 

12 

8 

u 

5  " 

13 

9  " 

5  "  14 

6  " 

6  " 

12 

7  " 

6  " 

13 

8 

u 

6  " 

14 

9  " 

6  "  15 

6  " 

7  " 

13 

7  " 

7  « 

14 

8 

it 

7  " 

15 

9  " 

7  "  16 

6  " 

8  " 

14 

7  " 

8  " 

15 

8 

u 

8  " 

16 

9  " 

8  "  17 

6  " 

9  " 

15 

7  " 

9  " 

16 

8 

u 

9  " 

17 

9  " 

9-  "  18 

6  " 

10  " 

16 

7  " 

10  " 

17 

8 

u 

10  " 

18 

9  " 

10  "  19 

6  " 

11  " 

17 

7  " 

11  " 

18 

8 

(( 

11  " 

19 

9  " 

11  "  20 

6  " 

12  " 

18 

7  " 

12  " 

19 

8 

u 

12  " 

20 

9  " 

12  "  21 

10  and 

0  are 

10 

11  and 

0  are 

11 

12 

and 

0  are 

12 

13  and 

0  are  13 

10  " 

1  " 

11 

11  " 

1  " 

12 

12 

a 

1  " 

13 

13  " 

1  "  14 

10  " 

2  " 

12 

11  " 

2  " 

13 

12 

u 

2  " 

14 

13  " 

2  "  15 

10  " 

3  " 

13 

11  " 

3  " 

14 

12 

ii 

3  " 

15 

13  " 

3  "  16 

10  " 

4  " 

14 

11  " 

4  " 

15 

12 

u 

4  " 

16 

13  " 

4  "  17 

10  " 

5  " 

15 

11  " 

5  " 

16 

12 

(( 

5  " 

17 

13  " 

5  "  18 

10  " 

6  " 

16 

11  " 

6  " 

17 

12 

II 

6  " 

18 

13  " 

6  "  19 

10  " 

7  " 

17 

11  " 

7  " 

18 

12 

{( 

7  " 

19 

13  " 

7  "  20 

10  " 

8  " 

18 

11  " 

8  " 

19 

12 

It 

8  " 

20 

13  " 

8  "  21 

10  " 

9  " 

19 

11  " 

9  " 

20 

12 

It 

9  " 

21 

13  " 

9  "  22 

10  " 

10  " 

20 

11  " 

10  " 

21 

12 

tl 

10  " 

22 

13  " 

10  "  23 

10  " 

11  " 

21 

11  " 

11  " 

22 

12 

tt 

11  " 

23 

13  " 

11  "  24 

10  " 

12  " 

22 

11  " 

12  " 

23 

12 

It 

12  " 

24 

13  " 

12  "  26 

1".  What  is  tho  process  called  by  which  we  find  the  sum  of  several  num- 
bers? 


ADDITION.  17 

2.  How  many  are  2  and  3  ?  2  and  5  ?  2  and  7  ?  2  and  9  ? 
2  and  4  ?  2  and  2  ?  2  and  8  ?  2  and  G  ? 

3.  How  many  are  3  and  3  ?  3  and  5  ?  3  and  7  ?  3  and 
9  ?  3  and  4  ?  3  and  6  ?  3  and  8  ?  3  and  3  ? 

4.  How  many  are  4  and  3  ?  4  and  5  ?  4  and  8  ?  4  and 
9  ?  4  and  1  ?  4  and  2  ?  4  and  4  ?  4  and  7  ? 

5.  How  many  are  5  and  3  ?  5  and  4  ?  5  and  7  ?  5  and 
8  ?  5  and  9  ?  5  and  2  ?  5  and  5  ?  5  and  6  ?  5  and  1  ? 

6.  How  many  are  6  and  2  ?  6  and  4  ?  6  and  3  ?  6  and  5  ? 

6  and  7  ?  6  and  9  ?  6  and  1  ?  6  and  6  ?  6  and  8  ? 

7.  How  many  are  7  and  3  ?  7  and  5  ?  7  and  7  ?  7  and  G  ? 

7  and  8  ?  7  and  9  ?  7  and  2  ?  7  and  4  ?  7  and  10  ? 

8.  How  many  are  8  and  2  ?  8  and  4  ?  8  and  5  ?  8  and  7  ? 

8  and  9  ?  8  and  8  ?  8  and  1  ?  8  and  3  ?  8  and  G  ? 

9.  Plow  many  are  9  and  1  ?  9  and  3  ?  9  and  5  ?  9  and  4  ? 

9  and  6  ?  9  and  8  ?  9  and  9  ?  9  and  2  ? 

10.  James  had  4  apples,  Samuel  gave  him  5  more,  and  John 
gave  him  G  ;  how  many  had  he  in  all  ? 

11.  Gave  7  dollars  for  a  bari-el  of  flour,  5  dollars  for  a  hun- 
dred weight  of  sugar,  and  8 -dollars  for  a  tub  of  butter;  what 
did  I  give  for  the  whole  ? 

12.  Paid  5  dollars  for  a  pair  of  boots,  12  dollars  for  a  coat, 
and  6  dollars  for  a  vest ;  what  was  the  whole  cost  ? 

13.  Gave  25  cents  for  an  arithmetic,  and  67  for  'a  geography  ; 
what  was  the  cost  of  both  ? 

TixusTRATiox.  —  25  equals  2  tens  and  5  units;  67  equals  6  tens 
and  7  units  ;  2  tens  and  6  tens  are  8  tens ;  and  5  units  and  7  units 
are  12  units,  or  1  ten  and  2  units  ;  1  ten  and  2  units  added  to  8  tens 
make  9  tens  and  2  units,  or  92.     Therefore  both  cost  92  cents. 

1 4.  On  the  fourth  of  July  20  cents  wore  given  to  Emily,  1 5 
cents  to  Betsey,  10  cents  to  Benjamin,  and  none  to  Lydia ; 
what  did  they  all  receive  ? 

15.  Bought  four  loads  of  hay ;  the  first  cost  15   dollars,  the 
second  12  dollars,  the  third  20  dollars,  and  the  fourth   17  dol 
lars  ;  what  was  the  price  of  the  whole  ? 

16.  Gave  55  dollars  for  a  horse,  40  dollars  for  a  wagon,  and 
17  dollars  for  a  harness ;  what  did  they  all  cost  ? 

17.  Sold  3  loads  of  wood  for  17  dollars,  6  tons  of  timber  for 
1 9  dollar's,  and  a  pair  of  oxen  for  GO  dollars ;  what  sum  did  I 
receive  ? 

2* 


18  ADDITION. 

18i  Addition  is  the  process  of  finding  the  sum  of  two  or  more 
numbers.     The  result  obtained  is  called  their  amount. 

Note. — Numbers  can  be  added  only  when  their  units  are  of  the  same 
kind. 

A  Sign  is  a  symbol  used  to  indicate  an  operation  to  be  per- 
formed. 

The  Sign  of  Addition  is  an  erect  cross,  -[-,  which  signifies  plus, 
or  added  to.  The  expression  7-J-5  is  read,  7  plus  5,  or  7  added 
to  5. 

The  Sign  of  Equality  is  two  parallel  horizontal  lines,  =,  and 

signifies  equal  to.     The  expression  7-|-5  =  12  is  read,  7  plus  5,  or 
7  added  to  5,  is  equal  to  12. 

19.  When  the  amount  of  each  column  is  less  than  10. 

Ex.  1.  A  man  bought  a  watch  for  42  dollars,  a  coat  for  16 
dollai's,  and  a  set  of  maps  for  21  dollars;  what  did  he  pay  for  the 
whole?  Ans.  79  dollars. 

OPERATION.  Having  arranged  the  numbers  so  that  all 

Dollars.  the  Units  of  the  same  order  shall  stand  in 

4  2  the  same  column,  we  first  add  the  columa 

1  g  of  units ;  thus,  1  and  6  are  7,  and  2  are  9 

2  \  (units),  and  write  down  the  amount  under 
the   column   of   units.     "We   then    add   the 

Amount  7t)  column  of  tens;  thus,  2  and  1  are  S,  and  4 

are  7  (tens),  which  we  write  under  the  col- 
umn of  tens,  and  thus  find  the  amount  of  the  whole  to  be  7t)  dollars. 

20.  First  Method  of  Proof.  —  Begin  at  the  top  and  add  the 
columns  downward  in  the  same  manner  as  they  were  added 
upward,  and  if  the  two  sums  agree  the  work  is  presumed  to  be 
right. 

By  adding  downward,  the  order  of  the  figures  is  inverted ; 
and,  therefore,  any  error  made  in  the  first  addition  Avould  prob- 
ably be  detected  in  the  second. 

This  method  of  proof  is  generally  used  in  business. 


18.  What  is  addition?  What  is  the  sign  of  addition,  and  what  does  it 
signify?  What  is  the  sign  of  equality,  and  what  does  it  signify  ? —  19.  How 
arc  numbers  arranged  for  addition?  AVhich  column  must  first  be  added  ? 
Why  ?  Wliere  do  you  place  its  sum  ?  Where  must  the  sum  of  each  column 
be  ])laccd  '.  What  is  tlu^  whole  sum  railed  ?  —  20.  How  is  addition  proved  J 
The  reason  for  this  method  of  jjroof?     Is  this  metiiod  in  common  use  ? 


ADDITION.  19 

Examples  for  Practice. 


2. 

3. 

4. 

5. 

Miles. 

Furlongs. 

Days. 

Weeks. 

151 

234 

472 

121 

212 

423 

315 

516 

321 

321 

102 

361 

Ans.    68  4 

6.  Wliat  is  the  sum  of  231,  114,  and  324?  Ans.  669. 

7.  Required  the  sum  of  235,  321,  and  142.  Ans.  698. 

8.  What  is  the  sum  of  11,  22,  505,  and  461  ?         Ans.  999. 

9.  Sold  twelve  plows  for  104  dollars,  two  wagons  for  214 
dollars,  and  one  chaise  for  121  dollars  ;  what  was  the  amount 
of  the  whole  ?  Ans.  439  dollars. 

10.  A  drover  bought  125  sheep  of  one  man,  432  of  another, 
and  of  a  third  311 ;  how  many  did  he  buy  ?      Ans.  868  sheep. 

21 1  When  the  sum  of  any  column  is  equal  to  or  ex- 
ceeds 10. 

Ex.  1.  I  have  three  lots  of  wild  land;  the  first  contains  246 
acres,  the  second  764  acres,  and  the  third  918  acres;  I  wish  to 
know  how  many  acres  are  in  the  three  lots.     Ans.  1928  acres. 

OPERATION.  Having  arranged  the  numbers  as  in 

Acres.  the  preceding  examples,  we  first  add 

2  4  6  the  units  ;  thus,  8  and  4  are  1 2,  and  6 

7  (3  4  are  18  (units),  equal  to  1  ten  and  8 

nig  units.     We  write  the   8  units  under 

the  column  of  imits,  and  carry  or  add 

Amount  19  2  8  *^^  ^  t®"^  *°  the  column  of  tens ;  thus, 

1  added  to  1  makes  2,  and  6  are  8, 
and  4  are  12  (tens),  equal  to  1  hundred  and  2  tens.  We  write  the  2 
tens  under  the  column  of  tens,  and  add  the  1  hundred  to  the  column 
of  hundreds;  thus,  1  added  to  9  makes  10,  and  7  are  17,  and  2  arc  19 
(hundreds),  ec^ual  to  1  thousand  and  9  hundreds.  We  write  the  9 
vuider  the  column  of  hundreds ;  and  there  being  no  other  column  to  be 
added,  we  set  down  the  1  thousand  in  thousands'  place,  and  find  the 
amount  of  the  several  numbers  to  be  1928. 

Note.  —  A  more  concise  way,  in  practice,  is  to  omit  calling  the  name  of 
each  tij^ure  as  added,  and  name  only  results. 

21.  When  the  sum  of  anycohimn  exceeds  ten,  where  are  the  units  written'? 
What  is  done  with  the  tens  ?  Why  do  you  carry,  or  add,  one  for  every  101 
How  is  the  sum  of  the  last  column  written  ? 


20 


ADDITION. 


22t  Rule.  —  Write  the  numbers  so  that  all  the  figures  of  the  same 
order  shall  stand  in  the  same  column. 

Add,  upward,  all  the  figures  in  the  column  of  units,  and,  if  the  amount 
be  less  than  ten,  write  it  underneath.  But,  if  the  amount  he  ten  or  mot-c, 
write  doion  the  unit  figure  only,  and  add  in  the  figure  denoting  the  ten  or 
tens  with  the  next  column. 

Proceed  in  like  manner  with  each  column,  until  all  are  added,  observ- 
ing to  write  down  the  whole  amount  of  the  last  column. 

23.  Second  Method  of  Proof.  —  Separate  the  numbers  to  be 
added,  if  there  are  more  than  two,  into  two  parts,  by  a  horizontal 
line.  Add  the  numbers  on  one  side  of  the  hne,  and  set  down 
their  sum.  Then  add  this  sum  and  the  remaining  number,  or 
numbers,  and,  if  their  sum  is  equal  to  the  first  amount,  the  work 
is  presumed  to  be  right. 

Is'OTE.  —  This  proof  depends  on  the  principle,  That  the  sum  of  all  the  parts 
vfaiiif  number  is  equal  to  its  whole. 

Examples   for   Practice. 


2. 

OPERATION. 
526 

317 
529 
132 

2. 

PROOF. 

520 

3. 

OPERATION. 
241 

532 
207 
9  13 

3. 

PROOF. 

241 

31  7 
529 
132 

532 
207 
913 

Ans.  15  0  4 


First  am't  15  0  4 


978 


Ans.    18  9  3 


First  am't  18  9  3 


1652 


Ans.  15  0  4 


Ans.  18  9  3 


4. 

5. 

6. 

7. 

8. 

9. 

Dollars. 

Miles. 

Pounds. 

Rods. 

Inches. 

Feet. 

1  1 

47 

127 

678 

789 

1769 

2  3 

87 

396 

97  1 

478 

7895 

97 

58 

787 

1  47 

71  9 

7563 

86 

83 

456 

71  6 

937 

8765 

217 

275 

17  66 

2512 

2923 

25992 

22.  Wliat  is  the  general  rule  for  addition  ?  —  23.  TMiat  is  the  second  method 
of  proving  addition  ?     Tiic  reusou  of  this  method  of  proof? 


ADDITION. 

lil 

10. 

11.                12. 

13. 

14. 

Ounces. 

Drams.                   Cents. 

Eagles. 

Degrees. 

876 

789             123 

471 

1234 

376 

567             478 

617 

3456 

7  1  5 

743             716 

871 

6544 

678 

435             478 

3  17 

7891 

910 

678             127 
16. 

899 

8766 

15. 

17. 

18. 

Feet. 

Inches. 

Hours. 

Minutes. 

7  8  9  5  6 

7  16  7  8 

71123 

98765 

37667 

12345 

45678 

12345 

12  3  4  5 

67890 

34680 

67111 

67890 

34567 

56777 

33333 

78999 

89012 

678  12 

71345 

13579 

789  17 

71444 

9  9  9  9  9 

19. 

20. 

21. 

22. 

Days- 

Years. 

Months. 

Hogsheads. 

17875897 

7  8  9567 

37 

30176 

7  16  7  5  12 

7613 

1378956 

3  1 

87  6567 

123  123 

70071  4 

8601 

98765 

7007  1 

367 

1  1 

7896 

475 

76117 

99  11 

789 

10  69 

4611779 

89120 

78 

374  176 

9171 

710 

7 

7  61 

1317  65 

4325 

23.  Add  1001,  76,  10078,  15,  8761,  7,  and  1678. 

Ans.  21616. 

24.  Add  49,  761,  3756,  8,  150,  761761,  and  18. 

Ans.  766503. 

25.  Eequired  the  sum  of  3717,  8,  7,  10001,  58,  18,  and  5. 

Ans.  13814. 

26.  Add  19,  181,  5,  897156,  81,  800,  and  71512. 

Ans.  969754. 

27.  What  is  the  sum  of  999,  8081,  9,  1567,  88,  91,  7,  and 
878?  Ans.  11720. 

28.  Add  71,  18765,  9111,  1471,  678,  9,  1446,  and  71. 

Ans.  31622. 

29.  Add  51,  1,  7671,  89,  871787,  61,  and  70001. 

Ans.  949661. 


22  ADDITION. 

30.  What  is  the  sum  of  71,  8956,  1,  785,  587,  and  76178  ? 

Ans.  86578. 

31.  Add  9999,  8008,  8,  81,  4777,  and  516785. 

Ans.  539658. 

32.  Add  5,  7,  8911,  467,  47895,  and  87.     Ans.  57372. 

33.  Add  123456,  71,  8005,  21,  and  716787.  Ans.  848340. 

34.  Add  47,  911111,  717,  81,  88767,  and  56. 

Ans.  1000779. 

35.  What  is  the  sum  of  71,  8899,  4,  7111,  and  678679  ? 

Ans.  694764. 

36.  Add  81,  879,  41,  76789,  42,  1,  and  78967. 

Ans.  156800. 

37.  Add  917658,  75,  876789,  46,  and  8222. 

Ans.  1802790. 

38.  Add  91,  76756895,  76,  14,  3,  and  76378. 

Ans.  76833457. 

39.  Add  10,  100,  1000,  10000,  100000,  and  1000000. 

Ans.  1111110. 

40.  What  is  the  sum  of  9,  99,  99,  1111,  8000,  and  5  ? 

Ans.  9323. 

41.  Add  41,  7651,  7678956,  43,  15,  and  6780. 

Ans.  7693486. 

42.  Add  1234,  7891,  3146751,  27,  9,  and  5. 

Ans.  3155917. 

43.  What  is  the  sum  of  19,  91,  1,  1,  1478,  1007,  and  46  ? 

Ans.  2643. 

44.  Add  four  hundred  seventy-six,  seventy-one,  one  hundred 
five,  and  three  hundred  eighty-seven.  Ans.  1039. 

45.  Add  fifty-six  thousand  seven  hundred  eighty-five,  seven 
hundred  five,  thirty-six,  one  hundi-ed  seventy  thousand  and  one, 
and  four  hundred  seven.  Ans.  227934. 

46.  Add  fifty-six  thousand  seven  hundred  eleven,  three  tliou- 
sand  seventy-one,  four  hundred  seventy-one,  sixty-one,  and  three 
tliousand  and  one.  Ans.  63315. 

47.  What  is  the  sum  of  the  folloAving  numbers  :  seven  hun- 
dred tliousand  seven  hundred  one,  seventeen  thousand  nine,  one 
milhon  six  hundred  thousand  seven  hundred  six,  forty-seven 
thousand  six  hundred  seventy-one,  seven  thousand  forty-se^■on, 
four  Inmdrcd  one,  and  nine?  Ans.  2373544. 

48.  Gave  73  dollar-;  for  a  watch,  15  dollars  for  a  cane,  119 
dollars  for  a  horse,  376  dollars  for  a  carriage,  and  7689  dollars 
for  a  house  ;  how  much  did  they  all  cost  ?     Ans.  8272  dollars. 


A-DDITION, 


23 


49.  In  an  orchard,  15  trees  bear  plums,  73  trees  bear  apples, 
29  trees  bear  pears,  and  14  trees  bear  cherries  ;  how  many  trees 
are  there  m  the  orchard?  Ans.  131  trees. 

50.  The  hind  quarters  of  an  ox  weighed  375  pounds  each,  the 
fore  quarters  315  pounds  each;  the  hide  weighed  96  pounds, 
and  the  tallow  87  pounds.  What  was  the  whole  weight  of  the 
ox?  Ans.  1563  pounds. 

51.  Bought  a  farm  for  1728  dollars,  and  sold  it  so  as  to  gain 
375  dollars  ;  how  much  was  it  sold  for?         Ans.  2103  dollars. 

52.  A  merchant  bought  five  pieces  of  cloth.  For  the  first  he 
gave  376  dollars,  for  the  second  198  dollars,  for  the  third  896 
dollars,  for  the  fourth  691  dollars,  and  for  the  fifth  96  dollars. 
How  much  did  he  give  for  the  whole?  Ans.  2257  dollars. 

53.  A  mei'chant  bought  five  hogsheads  of  molasses  for  375 
dollars,  and  sold  it  so  as  to  gain  25  dollars  on  each  hogshead; 
for  how  much  did  he  sell  it  ?  Ans.  500  dollars. 

54.  John  Smith's  farm  is  worth  7896  dollars;  he  has  bank 
stock  valued  at  369  dollars,  and  he  has  in  cash  850  dollars. 
How  miich  is  he  worth?  Ans.  9115  dollars. 

55.  Required  the  nimaber  of  inhabitants  in  the  New  England' 
States.  By  the  census  of  1850  there  were  in  Maine,  583,169; 
in  New  Hampshire,  317,976;  in  Massachusetts,  994,514;  in 
Rhode  Island,  147,545 ;  in  Connecticut,  370,792  ;  and  in  Ver- 
mont, 314,120.  Ans.  2,728,116. 

56.  Required  the  number  of  inhabitants  in  the  Middle  States, 
including  the  District  of  Columbia.  In  1850  there  were  in  New 
York,  3,097,394 ;  in  New  Jersey,  489,555  ;  in  Pennsylvania, 
2,311,786;  in  Delaware,  91,532 ;  in  Maryland,  583,034 ;  and  in 
the  District  of  Columbia,  51,687.  Ans.  6,624,988. 

57.  Required  the  number  of  inhabitants  in  the  Southern  States. 
In  1850  there  were  in  Virginia,  1,421,661;  in  North  CaroUna, 
869,039;  in  South  Carolina,  668,507;  in  Georgia,  906,185; 
and  in  Florida,  87,445.  Ans.  3,952,837. 

58.  Required  the  number  of  inhabitants  in  the  Southwestern 
States.  In  1850  there  were  in  Alabama,  771,623  ;  in  Mississippi, 
606,526  ;  in  Louisiana,  517,762  ;  in  Texas,  212^592  ;  in  Arkansas, 
209,897  ;  and  in  Tennessee,  1,002,917.  Ans.  3,321,317. 

59.  Required  the  number  of  inhabitants  in  the  Northwestern 
States  and  Territories.  In"  1850  there  were  in  Missouri,  682,044; 
in  Kentucky,  982,405  ;  in  Ohio,  1,980,329  ;  in  Indiana,  988,416; 
in    lUinois,    851,470 ;    in    Michigan,    397,654 ;    in    Wisconsin, 


24 


ADDITION. 


305,391;  in  Iowa,  192,214:;  in  California,  92,597;  and  in  the 
Territories,  92,298.  Ans.  6,564,818. 

24.  To  add  two  or  more  columns  at  a  single  opera- 
tion. 

Ek.  1.  "Washington  lived  68  years;  John  Adams,  91  years; 
Jefferson,  83  years ;  Madison,  85  years.  What  is  the  sum  of 
the  years  they  all  Uved  ?  Ans.  327. 


Beginning  with  the  number  last  -vrritteD 
down,  we  add  the  units  and  tens,  thus  :  85 
and  3  equal  88,  and  80  equal  1G8,  and  1 
equal  169,  and  90  equal  259,  and  8  equal 
267,  and  60  equal  327,  the  sum  sought.  In 
like  manner  may  be  added  more  than  two 
columns  at  one  operation. 


Note.  —  The  examples  that  follow  can  be  performed  as  the  above,  or  by 
the  common  method,  or  by  both,  as  the  teacher  may  advise. 


OPERATIO:^f. 

Years. 

68 

91 

83 

85 

;  327 

4. 


6. 


Ounces. 

Yards. 

Feet. 

Inches. 

Chaldrons. 

1234 

2345 

3456 

7891 

5678 

5678 

6789 

7891 

135  6 

32  15 

9012 

10  2  3 

3456 

7891 

6789 

3456 

445  6 

7891 

2345 

3214 

7890 

7890 

345  6 

6789 

1234 

13  45 

1234 

7890 

1234 

3789 

6789 

5678 

1378 

5678 

1379 

3216 

9012 

8  1  23 

9  123 

9008 

7890 

3456 

4567 

4567 

1071 

1030 

7890 

89  12 

89  12 

71  63 

7055 

1345 

3456 

3456 

6781 

5  67  8 

6789 

7891 

7812 

1780 

1234 

3  4  5  6 

3456 

3  45  6 

3007 

5678 

7890 

7891 

7812 

5617 

9001 

5678 

3783 

3713 

4456 

2345 

9012 

1237 

7891 

3456 

6789 

3  45  6 

789  1 

135  7 

789  1 

1030 

7890 

1007  • 

9  0  0  9 

3070 

781  6 

1234 

5670 

8  7  65 

4567 

1781 

5678 

1234 

4321 

3456 

SUBTRACTION. 


25 


SUBTRACTION. 

25.  "When  it  is  required  to  find  the  difference  between  two 
numbers  of  the  same  khid,  the  process  is  called  SubtrslCtiOll.  The 
operation  is  the  reverse  of  addition. 

Ex.  1.  John  has  7  oranges,  and  his  sister  but  4 ;  how  many 
more  has  John  than  his  sister  ? 

iLLUSTRATrox.  —  We  first  inquire  what  number  added  to  4  will 
make  7.  From  addition  Ave  learn  that  4  and  3  are  7  ;  consequently, 
if  4  oranges  be  taken  from  7  oranges,  3  oranges  will  remain.  Hence 
John  has  3  oranges  more  than  his  sister. 


SUBTRACTION  TABLE. 


1  from     1  leaves  0 

2  from    2  leaves  0 

3  from    3  leaves  0 

4  from    4  leaves  0 

1    "       2     "      1 

2    •'       3      "      1 

3    •'       4     "      1 

4    "       5      "      1 

1    "       3     "      2 

•  2    "       4     "      2 

3    "       5     "      2 

4     "       6      "      2 

1    "       4     "      3 

2    "       5      "      3 

3    "       6     "      3 

4    "       7      "      3 

1    "       5     "      4 

2    "       6     "      4 

3    "       7     "      4 

4    "       8     "      4 

1    "       G     "      5 

2    "       7      "      5 

3    "       8     "      5 

4    "       9     "      5 

1    "       7     "      6 

2    "       8     "      6 

3    "       9     "      6 

4    "     10     "      6 

1    "       8     "      7 

2    "       9      "      7 

3    "     10     "      7 

4    "     11      «      7 

1    "       9     "      8 

2    "     10      "      8 

3    "     11      "      8 

4    "     12      "      8 

1    "     10     "     ^ 

2    "      11      "      9 

3    "     12     "      9 

4    "     13     "      9 

1    "     11     "    10 

2    "     12      "    10 

3    "      13     "    10 

4    "     14     "    10 

1    "     12     "    11 

2    "     13     "    11 

3    "      14     "    11 

4    "     15     "    11 

1    «     13     «    12 

2    "      14      "    12 

3    "     15     "    12 

4    "      16     "    12 

5  from    5  leaves  0 

6  from    6  leaves  0 

7  from    7  leaves  0 

8  from    8  leaves  0 

5    "       6     "      1 

6    "        7      "       1 

7    "       8     "      1 

8    "       9     "      1 

5    "       7     "      2 

6    "        8      "      2 

7    "       9     "      2 

8    "     10     "      2 

5    "       8     "      3 

6    "       9     "      3 

7    "     10     "      3 

8    "     11      «      3 

5    "       9     "      4 

6    "      10     "      4 

7    "     11     "      4 

8    "      12     "      4 

5    "     10     "      5 

6    «     11      "      5 

7    "     12     "      5 

8    "     13     "      5 

5    "     11     "      6 

6    "     12     "      6 

7    "     13     "      6 

8    "     14     "      6 

5    "     12     '.'      7 

6    "      13     "      7 

7    "     14     "      7 

8    "     15     "      7 

6    «     13     "      8 

6    "      14     "      8 

7    "     15     "      8 

8    "     16     »      8 

6    "     14     "      9 

6    "      15     "      9 

7    "     16     "      9 

8    "     17     "      9 

5    "     15     "    10 

6    "      16     "    10 

7    "     17     "    10 

8    "      18      "    10 

6    "     16     "    11 

6    "     17     "    11 

7    "     18     "    11 

8    »     19      «    11 

6    "     17     "    12 

6    "      18     "    12 

7    "     19     "    12 

8    "     20      "    12 

9  from    9  leaves  0 

10  from  10  leaves  0 

11  from  11  leaves  0 

12  from  12  leaves  0 

9    "     10     "      1 

10    '•      11      "      1 

11    "      12     "      1 

12    "     13     "      1 

9    "     11     "      2 

10    "      12     "      2 

11    "     13     "      2 

12    "     14     "      2 

9    "     12     "      3 

10    "      13     «      3 

11    "     14     "      3 

12    "     15     "      8 

9    "     13     "      4 

10    "      14     "      4 

11    "     15     "      4 

12    "     16     "      4 

9    «     14     «      5 

10    «     15     "      5 

11    "     16     "      5 

12    "     17     "      6 

9    "     15     "      6 

10    "      16     "      6 

11    "     17     "      6 

12    "     18     "      6 

9    "     IS     "      7 

10    «     17     "      7 

11    «     18     "      7 

12    "     19     '•      7 

9    "     17     "      8 

10    "      18     "      8 

11    "      19     "      8 

12    "     20     "      8 

9    "     13     "      9 

10    "      19     "      9 

11    "     20     "      9 

12    "     21     "      » 

9    "     19     "    10 

10    "     20     "    10 

11    "     21     "    n 

12    "     22     "    10 

9    »     20     "    11 

W    "     21     "    11 

11    "     22     "    11 

12    "     23     "    11 

9    "     21     "    12 

10    "     22     "    12 

11    "     23     "    12 

12    "     24     "    12 

25.  What  does  subtraction  teach  1     Of  what  is  it  the  reverse  1 


26  SUBTRACTION. 

2.  Thomas  Lad  five  orauges,  and  gave  two  of  tliem  to  Jolm ; 
liow  many  had  he  left  ? 

3.  Peter  had  six  marbles,  and  gave  two  of  them  to  Samuel ; 
how  many  had  he  left  ? 

4.  Lydia  had  four  cakes ;  having  lost  one,  how  many  had  she 
left? 

5.  Daniel,  having  eight  cents,  gives  three  to  Mary ;  how  many 
has  he  left  ? 

6.  Benjamin  had  ten  nuts  ;  he  gave  four  to  Jane,  and  three  to 
Emily ;  how  many  had  he  left  ? 

7.  Moses  gives  eleven  oranges  to  John,  and  eight  to  Enoch ; 
how  many  more  has  John  than  Enoch  ? 

8.  Paid  seven  dollars  for  a  pair  of  boots,  and  two  dollars  for 
shoes  ;  how  much  did  the  boots  cost  more  than  the  shoes  ? 

9.  How  many  are  4  less  2  ?     4  less  1  ?    4  less  4  ? 

10.  How  many  are  4  less  3  ?     5  less  1  ?     5  less  5  ? 

11.  How  many  are  5  less  2  ?     5  less  3  ?     5  less  4  ? 

12.  How  many  are  6  less  1  ?     6  less  2  ?     6  less  4  ?    6  less  5  ? 

13.  How  many  are  7  less  2  ?     7  less  3  ?     7  less  4  ?    7  less  6  ? 

14.  How  many  are  8  less  6  ?     8  less  5  ?     8  less  2  ?    8  less  4  ? 

15.  How  many  are  9  less  2  ?     9  less  4  ?    9  less  5  ?     9  less  7  ? 

16.  How  many  are  10  less  8  ?     10  less  7  ?     10  less  5  ?     10 
less  3  ?     10  less  1  ? 

17.  How  many  are  11  less  9  ?     11  less  7  ?     11  less  5  ?     11 
less  3?     11  less  4? 

18.  How  many  are  12  less  10  ?     12  less  8  ?     12  less  6  ?     12 
less  4?     12  less  7? 

19.  How  many  are  13  less  11  ?    13  less  10  ?     13  less  7  ?    13 
less  9  ?     13  less  5  ? 

20.  How  many  are  14  less  11  ?     14  less  9  .^     14  less  8  ?     14 
less  6  ?     14  less  7  ?     14  less  3  ? 

21.  How  many  arc  15  less  2  ?     15  less  4?     15  less  5  ?     15 
less  7  ?     15  less  9  ?     15  less  13  ? 

22.  How  many  are  16  less  3?     16  less  4?     16  less  7?     16 
less  9?     16  less  11?     16  less  15? 

23.  How  many  are  17  less  1  ?     17  less  3  ?     17  less  5  ?     17 
less  7  ?     17  less  8  ?     17  less  12  ? 

24.  How  many  are  18  less  2  ?     18  less  4?     18  less  7  ?     18 
less  8?     18  less  10?     18  less  12? 

25.  How  many  are  19  less  1  ?     19  less  3?     19  less  5?     19 
less  7?     19  less  9?     19  less  16? 


SUBTRACTION.  27 

26.  How  many  are  20  less  5?  20  less  8?  20  less  9?  20 
less  12?     20  less  15?     20  less  19  ? 

27.  Bought  a  horse  for  GO  dollars,  and  sold  him  for  90  dol- 
lars ;  how  much  did  I  gain  ? 

Illustration.  —  60  equals  6  tens,  and  90  equals  9  tens ;  6  tens 
from  9  tens  leave  3  tens,  or  30.     Thei'efore  I  gained  30  doUai-s. 

28.  Sold  a  wagon  for  70  dollars,  which  cost  me  100  dollars ; 
how  much  did  I  lose  ? 

29.  John  travels  30  miles  a  day,  and  Samuel  90  miles  ;  what 
is  the  difference  ? 

30.  I  have  100  dollars,  and  after  I  shall  have  given  20  to 
Benjamin,  and  paid  a  debt  of  30  dollars  to  J.  Smith,  how  many 
dollars  have  I  left  ? 

31.  John  Smith,  Jr.,  had  170  dollars ;  he  gave  his  oldest 
daughter,  Angeline,  40  dollars,  his  youngest  daughter,  Mary,  50 
dollars,  his  oldest  son,  James,  30,  and  his  youngest  son,  William, 
20  dollars ;  he  also  paid  20  dollars  for  his  taxes ;  how  many 
dollars  had  he  remaining  ? 


'o 


26.  Subtraction  is  the  taking  of  one  number  from  another  to 
find  the  difference. 

The  Minuend  is  the  number  to  be  diminished. 

The  Subtraliend  is  the  number  to  be  subtracted. 

The  Difference,  or  Remailtder,  is  the  result,  or  the  number  left 
after  Subtraction. 

Note  1.  —  One  number  can  bo  subtracted  from  another  only  when  the 
units  of  both  are  of  the  same  kind. 

Note  2.  —  When  the  two  given  numbers  are  unequal,  the  greater  number 
is  the  minuend. 

27.  Sign  of  Subtraction  is  a  short  horizontal  line,  thus  — , 
signifying  minus,  or  less.  It  indicates  that  the  number  following 
is  to  be  taken  from  the  one  that  precedes  it.  The  expi'ession 
6  —  2  =  4  is  read,  6  minus,  or  less,  2  is  equal  to  4. 

26.  What  is  subtraction  1  What  is  the  greater  number  called  ?  What  is 
the  less  number  called  ?  Wliat  the  answer  1  — 27.  The  sign  of  subtraction  ? 
What  does  it  signify  and  indicate  1 


28 


SUBTRACTION. 


28.   Wlien  each  figure  in  the  subtrahend  is  less  than 
the  figure  above  it  in  the  minuend. 

Ex.  1.     Let  it  be  required  to  take  245  from  468,  and  to  find 
their  difference.  ^  Ans.  223. 

OPERATION. 


Minuend      4  6  8 
Subtrahend  2  4  5 

Remainder  2  2  3 


We  place  the  less  number  under  the 
greater,  units  under  units,  tens  under  tens, 
&c.,  and  draw  a  line  below  them.  We  then 
begin  at  the  right,  and  say,  5  units  from  8 
units  leave  3  units,  and  write  the  3  in  units' 
place  below  ;  then  4  tens  from  6  tens  leave 
2  tens,  and  write  the  2  in  tens'  place  below  ;  and  2  hundreds  from  4 
hundreds  leave  2  hundreds,  which  we  write  in  hundreds'  place  below ; 
and  thus  find  the  difference  to  be  223. 

29t  First  Method  of  Proof .  —  Add  the  remainder  and  the  sub- 
trahend together,  and  their  sum  will  be  equal  to  the  minuend,  if 
the  work  is  right. 

Note.  —  This  proof  depends  on  the  principle,  "That  the  greater  of  any  two 
numbers  is  equal  to  tlie  less  added  to  their  difference. 


Examples  for  Practice. 


OPERATION. 

Minuend       5  4  7 
Subtrahend  2  35 


Remainder    312 


2. 

PROOF. 

547 
235 

312 


Min.  5  4  7 


3. 

OPERATION. 

986 
763 


223 


3. 

PROOF. 

986 
7  63 

223 


Min.  9  8  6 


From 
Take 


4. 

684 

4  6  2. 


5. 

735 
523 


864 
651 


948 
746 


8.  A  farmer  paid  539  dollars  for  a  span  of  fine  horses,  and 
sold  them  for  425  dollars  ;  how  much  did  he  lo>e  ? 

Ans.  114  dollars. 

9.  A  farmer  raised  896  bushels  of  wheat,  and  sold  675  bushels 
of  it ;  how  much  did  he  reserve?  Ans.  221  bushels. 


28.  How  .ire  numbers  arranjjrd  for  suhtr.icfion  ?  "Whore  do  yon  hejrin  to 
Bubtract?  Why?  Where  do  you  write  the  difference  ?  —  29.  What  is  tho 
first  method  of  provitifr  subtraction  ?  Tlic  reason  for  this  proof,  or  on  what 
principle  docs  it  depend  ? 


SUBTRACTION.  29 

10.  A  gentleman  gave  his  son  3692  dollars,  and  his  daughter 
1212  dollars  less  than  his  son;  how  much  did  he  give  his 
daughter  ?  Ans.  2480  dollars- 

30.  When  any  figure  in  the  subtrahend  is  greater 
than  the  figure  above  it  in  the  minuend. 

Ex.  1.  If  I  have  624  dollars,  and  lose  342  of  them,  how 
many  remain  ?  Ans.  282. 

OPERATION.  ^^^^  fii'st  take  the  2  units  from  the  4  units, 

TVr'niipnd       6  2  4  ^^^  ^"'^  ^^'^  difference  to  be  2  units,  which 

G  1  f     I       1^/19  ^®  write  below.     We  then  proceed  to  take 

bubtraliend  o  4  ^  ^1^^  ^  ^^j^^  ^^,^^  ^^^  2  tens  above  it ;  but  we 

Kemainder  2  8  2  J^f  ^«  ^^^  .^  difficulty,  since  the  4  is  greater 

than  the  2,  and  cannot  be  subtracted  from 
it.  We  therefore  add  10  to  the  2,  which  makes  12  tens,  and  then  sub- 
tract the  4  from  the  12,  and  8  tens  remain,  which  we  write  below. 
Then,  to  compensate  for  the  10  thus  added  to  the  2  in  the  minuend, 
we  add  1  to  the  3  in  the  next  higher  place  in  the  subtrahend,  which 
makes  4  hundreds,  and  subtract  the  4  tiom  the  6,  and  2  hundreds  re- 
main ;  and  thus  find  the  remainder  to  be  282. 

Note  1. — This  operation  depends  upon  the  self-evident  truth,  That,  if 
anij  tico  numbers  are  equallij  incnuseU,  .their  Jijjhejice  remains  the  same.  In 
working  the  example  10  tens,  equal  to  1  Imndred,  were  added  to  the  2 
tens  in  the  upper  number,  and  1  was  added  to  the  3  hundreds  in  the  lower 
nnniber.  Now,  since  the  3  stands  in  the  hundreds'  place,  the  1  added  was 
in  fact  1  hundred.  Hence,  the  two  numbers  being  equally  increased,  the 
difference  is  the  same. 

Note  2.  — In  the  operation,  instead  of  adding:  10  to  the  2  in  the  minuend, 
1  of  tlie  6  hundreds  can  be  joined  to  tlie  2  tens,  thus  forming  12  tens;  then 
4  tens  from  12  tens  leaves  8  tens  ;  and  1  of  the  6  hundreds  having  been  taken, 
there  remain  only  5  hundreds ;  and  3  hundreds  from  .5  hundreds  leaves  2 
hundreds,  and  the  result  is  the  same  as  by  tiie  other  process. 

31 1  EuLE.  —  Place  the  less  number  under  the  greater,  so  that  figures 
of  the  same  order  shall  stand  in  the  same  column. 

Commencing  at  the  right  hand,  subtract  each  figure  of  the  subtrahend 
from  the  figure  above  it. 

If  any  figure  of  the  subtrahend  is  larger  than  the  figure  above  it  in  the 
minuend,  add  \0  to  that  figure  of  the  minuend  before  subtracting,  and 
then  add  1  to  the  next  figure  of  the  subtrahend. 

30.  How  do  you  proceed  when  a  figure  of  the  subtrahend  is  larger  than 
the  one  above  it  in  the  minuend  ?  How  do  you  compensate  for  the  10  wliieh 
is  added  to  the  minuend  ?  The  reason  for  this  addition  to  the  minuend  and 
subtrahend  1  How  does  the  1  added  to  the  subtrahend  equal  the  10  added 
to  the  minuend  ?  —  31.  The  rule  for  subtraction  ] 
3* 


30 


SUBTRACTION. 


32i  Second  Method  of  Proof.  —  Subtract  the  remainder  or 
difference  from  tlie  minuend,  and  the  result  will  be  like  the 
subtrahend,  if  the  work  is  right. 

Note.  —  This  proof  depends  on  the  principle,  That  the  smaller  of  any  two 
numbers  is  equal  to  tfie  larger  'less  their  difference. 

Examples  for  Practice. 


OPERATION. 

PROOF. 

OPERATION. 

PROOF. 

Minuend       3  7  6 

37  6 

531 

531 

Subtraliend  1  6  7 

167 

389 

389 

Remainder  2  0  9 


209 


Sub.     16  7 


142 


1  42 


Sub.     3  8  9 


4. 


6. 


Tons. 

Gallons. 

Pecks. 

Feet. 

From 

978 

67158 

14711 

100000 

Take 

199 
779 

14339 

9  197 

90909 

Ans. 

52819 

5514 

9091 

8. 

9. 

10. 

11. 

Miles. 

Dollars. 

Minutes. 

Seconds, 

From 

67895 

456798 

765321          555555 

Take 

19999 

190899 

17777 

7          177777 

12. 

13. 

Rods. 

Acres. 

From 

100200300400500 

1000000000000 

Take 

908070G0504030 
From  671111  take  199999. 

999999999999 

14. 

Ans.  471112. 

15. 

From  1789100  take  808088. 

Ans.  981012. 

16. 

From  1000000  take  999999. 

Ans.  1. 

17. 

From  9999999  take  1607. 

Ans.  9998392. 

18. 

From  610] 

L507601061  take  3806790989 

• 

Ans.  6097700810072. 

32.  Wliat  is  the  second  mctliod  of  provinjj  subtraction  1     "What  is  the 
reason  for  this  method  of  proof,  or  on  what  principle  docs  it  depend  1 


SUBTRACTION.  31 

19.  From  8054010657811  take  7G909748598. 

Alls.  7977100909213. 

20.  From  7100071G41115  take  10071178. 

Ans.  7100061569937. 

21.  From  501505010678  take  794090589. 

Ans.  500710920089. 

22.  Take  99999999  from  100000000.  Ans.  1. 

23.  Take  44444444  from  500000000.    Ans.  455555556. 

24.  Take  1234567890  from  9987654321. 

Ans.  8753086431. 

25.  From  800700567  take  1010101.  Ans.  799690466. 

26.  Take  twenty-five  thousand  twenty-five  from  twenty-five 
millions.  Ans.  24974975. 

27.  Take  nine  thousand  ninety-nine  from  ninety-nine  thou- 
sand. Ans.  89901. 

28.  From  one  hundred  one  millions  ten  thousand  one  hundred 
one  take  ten  millions  one  hundred  one  thousand  and  ten. 

Ans.  90909091. 

29.  From  one  million  take  nine.  Ans.  999991. 

30.  From  three  thousand  take  thirty-three.  Ans.  2967. 

31.  From  one  hundred  millions  take  five  thousand. 

Ans.  99995000. 

32.  From  1,728  dollars,  I  paid  961  dollars  ;  how  many  re- 
main ?  Ans.^  767  doUars. 

33.  Our  national  independence  was  declared  in  1776  ;  how 
many  years  from  that  period  to  the  close  of  the  last  war  with 
Great  Britain,  in  1815  ?  Ans.  39  years. 

34.  The  last  transit  of  Venus  was  in  1769,  and  the  next 
will  be  in  1874;  how  many  years  will  intervene? 

Ans.  105  years. 

35.  The  State  of  New  Jersey  contains  6851  square  miles,  and 
Delaware  2120.  How  many  more  squai'e  miles  has  the  former 
State  than  the  latter  ?  Ans.   4731. 

36.  In  1840  the  number  of  inhabitants  in  the  Tlnited  States 
was  17,069,453,  and  in  1850  it  was  23,191,876  ;  what  was  the 
increase?  Ans.  6,122,423, 

37.  In  1850  there  were  i-aised  in  the  State  of  Ohio  56,619,608 
bushels  of  corn,  and  in  1853,  73,436,690  bushels  ;  what  was  the 
increase?  Ans.  16,817,082  bushels. 

38.  By  the  census  of  1850,  13,121,498  bushels  of  wheat  were 
raised  in  New  York,  and  15,367,691  bushels  in  Pennsylvania; 
how  many  bushels  in  the  latter  State  more  than  in  the  former  ? 

Ans.  2,246,193  bushels. 


93 


SUBTRACTION. 


39.  The  city  of  New  York  owes  13,960,856  dollars,  and 
Boston  owes  7,779,855  dollars ;  how  much  more  does  2sew  York 
owe  than  Boston  ?  Ans.  6,181,001  dollars. 

40.  From  five  hundred  eighty-one  thousand  take  three  thou- 
sand and  ninety-six.  Ans.  577,904. 

41.  It  was  ascertained  by  a  transit  of  Venus,  June  3,  1769, 
that  the  mean  distance  of  the  earth  from  the  sun  is  ninetv-five 
millions  one  hundred  seventy-three  thousand  one  hundred  twenty- 
seven  miles,  and  that  the  mean  distance  of  Mars  from  the  sun 
is  one  hundred  forty-five  millions  fourteen  thousand  one  hundred 
forty-eight  miles.     Required  the  difierence  of  the  di>tances. 

Ans.  49,841,021  miles. 

33.  To  subtract  when  there  are  two  or  more  subtra- 
hends. 

Ex.  1.  A  man  owing  767  dollars,  paid  at  one  time  190  dollars, 
at  another  time  131  dollars,  at  another  time  155  dollars  ;  how 
much  did  he  then  owe  .''  Ans.  291  dollars. 

In  the  first  opera- 
tion, the  several  sub- 
trahends, lyiving  been 
properly  plaoed,  are 
added  for  a  sinf/le  sub- 
trahend, to  be  taken 
from  the  minuend. 
In  the  second,  the  sub- 
trahends are  substraci- 
ed,  as  they  are  added, 
at  one  operation,  thus: 
beginning  with  units, 


Minuend 


FIRST  OPERATION. 
Dollars. 

767 


SECOXD   OPERATION. 
Dollars. 

Minuend  7  6  7 


131 

190 
15^ 


f  131 

Subtrahends  ^19  0 

(155 


Subtrahend  4  7  6    Remainder       2  9  1 


Remainder  2  9  1 


5  and  1  equal  6,  which  from  7  units  leaves  1  unit ;  passing  to  tens,  5 
and  9  and  3  equal  1 7  tens  ;  reserving  the  left-hand  figure  to  add  in 
with  the  figures  of  the  subtrahends  in  the  next  column,  the  right-hand 
figure,  7,  being  larger  than  C  tens  of  the  minuend  we  add  10  to  the  6, 
and,  subtracting,  have  left  9  tens  ;  and,  passing  to  hundreds,  we  add 
in  the  left-hand  figure  1  reserved  from  tlie  17  tens,  and  also  add  1, 
equal  10  tens,  to  compensate  for  the  10  added  to  the  minuend,  and 
with  the  other  figures,  1  and  1  and  1  equal  5  hundreds,  which,  taken 
from  7  hundreds,  leave  2  hundreds;  and  291  as  the  answer. 

2.  E.  "Webster  owned  6,765  acres  of  land,  and  he  gave  to  his 
oldest  brother  2,196  acres,  and  his  uncle  Rollins  1,981  acres; 
how  much  has  he  left  ?  Ans.  2,588  acres. 

3.  The  real  estate  of  James  Dow  is  valued  at  3,769  dollars, 
and  his  personal  estate  at  2,648  dollars  ;  he  owes  John  Smith 
1,728  dollars,  and  Job  Tyler  1,161  dollars  ;  how  much  is  he 
worth  ?  Ans.  3,528  dollars. 


MULTIPLICATION. 


33 


MULTIPLICATION. 

34.  When  any  number  is  to  be  added  to  itself  several  times, 
the  operation  may  be  shortened  by  a  process  called  Multiplicalion. 

Ex.  1.  If  a  man  can  earn  8  dollai's  in  1  week,  what  will  he 
earn  in  4  weeks  ? 

Illustration.  —  It  is  evident,  since  a  man  can  earn  8  dollars  in  1 
week,  that  in  4  weeks  he  will  earn  4  times  as  much,  and  the  result 
may  be  obtained  by  addition  ;  thus,  8-j-8-f-8-|-8  =  32  dollars ; 
or,  by  a  more  convenient  process,  by  multiplying  by  4,  the  number  of 
times  8  dollars  is  to  be  taken;  thus,  4  times  8  dollars  are  32  dollars. 
Hence  in  4  weeks  he  will  earn  32  dollars. 


MULTIPLICATION    TABLE. 


2  times   1 

a.ve 

2 

3  times    1 

are 

3 

4 

times 

1 

ars 

4 

5  times 

1 

are 

5 

2      "      2 

u 

4 

3      "      2 

11 

6 

4 

2 

it 

8 

5 

2 

'• 

10 

2      "      3 

C( 

6 

3      "      3 

11 

9 

4 

3 

It 

12 

5 

3 

It 

15 

2      ''      4 

(( 

8 

3      "      4 

11 

12 

4 

4 

11 

16 

5 

4 

11 

20 

2      "      5 

(( 

10 

3      "      5 

11 

15 

4 

5 

11 

20 

5 

5 

11 

25 

2      "      6 

(( 

12 

3      "      6 

11 

18 

4 

6 

it 

24 

5 

6 

11 

30 

2      "      7 

l£ 

14 

3      "      7 

11 

21 

4 

7 

it 

28 

5 

7 

It 

35 

2      "      8 

U 

1(5 

3      "      8 

11 

24 

4 

8 

it 

32 

5 

8 

It 

40 

2      "      9 

a 

18 

3      "      9 

11 

27 

4 

9 

11 

36 

5 

9 

It 

45 

2      "    10 

a 

20 

3      "    10 

11 

30 

4 

10 

11 

40 

5 

10 

It 

50 

2      »    11 

u 

22 

3      "    11 

11 

33 

4 

11 

11 

44 

5 

11 

11 

55 

2      "    12 

u 

24 

3      "    12 

11 

36 

4 

12 

11 

48 

5 

12 

11 

60 

6  times   1 

are 

6 

7  times   1 

are 

7 

8  times 

1 

are 

8 

9  times 

1 

are 

9 

6      "      2 

u 

12 

7      "      2 

11 

14 

8 

2 

11 

16 

9 

2 

it 

18 

6      "      3 

it 

18 

7      "      3 

11 

21 

8 

3 

It 

24 

9 

3 

It 

27 

6      "      4 

(I 

24 

7      "      4 

11 

28 

8 

4 

It 

32 

9 

4 

It 

36 

6      "      5 

CI 

30 

7      "      6 

11 

35 

8 

5 

11 

40 

9 

5 

It 

45 

6      "      6 

t( 

30 

7      "  •   6 

11 

42 

8 

6 

11 

48 

9 

6 

It 

54 

6      "      7 

C( 

42 

7      "      7 

11 

49 

8 

7 

u 

56 

9 

7 

It 

63 

6      "      8 

11 

48 

7      "      8 

11 

56 

8 

8 

ti 

64 

9 

8 

11 

72 

6      "      9 

it 

54 

7      "      9 

it 

63 

8 

9 

11 

72 

9 

9 

It 

81 

6      "    10 

C( 

60 

7      "    10 

11 

70 

8 

10 

It 

80 

9 

10 

It 

90 

6      "    11 

11 

6t3 

7      "    11 

11 

77 

8 

11 

It 

88 

9 

11 

11 

99 

6      "    12 

11 

72 

7      "    12 

11 

84 

8 

12 

It 

93 

9 

12 

It 

108 

10  times   1 

are 

10 

10  times  11 

are 

110 

11  times   7 

are 

77 

12  timet 

3 

are 

36 

10      "      2 

11 

20 

10      "     12 

11 

120 

11 

8 

It 

88 

12 

4 

11 

48 

10      "      3 
10      "      4 

11 

It 

30 
40 

11 
11 

9 
10 

11 
11 

99 
110 

12 
12 

5 
6 

It 
It 

60 
72 

10      "      5 

11 

50 

11  times   1 

are 

11 

11 

11 

It 

121 

12 

7 

It 

84 

10      "      6 

11 

60 

11      "      2 

It 

22 

11 

12 

11 

132 

12 

8 

It 

96 

10      "      7 
10      "      8 

11 
11 

70 
80 

11      "      3 
11      "      4 

11 
11 

33 
44 

12 
12 

9 

10 

It 
11 

108 
120 

10      "      9 

11 

90 

11      "      5 

11 

55 

12 

times   1 

are 

12 

12 

11 

ti 

132 

10      "    10 

11 

100 

11      "      6 

It 

66 

12 

11 

2 

it 

24 

12 

12 

It 

144 

34.  Bow  may  the  process  of  adding  a  number  to  itself  several  times  be 
shortened  ■? 


84  MULTIPLICATION. 

2.  What  cost  5  barrels  of  flour  at  6  dollars  per  barrel  ? 

Illustratiox.  —  If  1  barrel  of  flour  cost  6  dollars,  5  barrels  will 
cost  5  times  as  much  ;  5  times  6  dollars  are  30  dollars.  Tlieretbre  5 
barrels  of  flour  at  6  dollars  per  barrel  will  cost  30  doUai-s. 

3.  What  cost  6  bushels  pf  beans  at  2  dollars  per  bushel  ? 

4.  What  cost  5  quarts  of  cherries  at  7  cents  per  quart  ? 

5.  What  will  7  quarts  of  vinegar  cost  at  12  cents  per  quart? 

6.  What  cost  9  acres  of  land  at  10  dollars  per  acre. 

7.  If  a  pint  of  currants  cost  4  cents,  what  cost  9  pints  ? 

8.  If  in  one  penny  there  are  4  farthings,  how  many  in  9  pence  ? 
In  7  pence  ?     In  8  pence  ?     In  4  pence  ?     In  3  pence  ? 

9.  If  12  pence  make  a  shilling,  how  many  pence  in  3  shil- 
linsrs  ?     In  5  shillings  ?     In  seven  shillinofs  ?     In  9  shillings  ? 

10.  If  one  pound  of  raisins  cost  6  cents,  what  cost  4  pounds  ? 
5  pounds  ?  6  pounds  ?  7  pounds  ?  8  pounds  ?  9  pounds  ? 
10  pounds  ?     12  pounds  ? 

11.  In  one  acre  there  are  four  roods  ;  how  many  roods  in  2 
aci'es  ?  In  3  acres  ?  In  4  acres  ?  In  5  acres  ?  In  6  aci'es  ? 
In  9  acres  ? 

12.  A  good  pair  of  boots  is  worth  5  dollars  ;  what  must  I  give 
for  5  pairs  ?  For  G  pairs  ?  For  7  pairs  ?  For  8  pairs  ?  For 
9  pairs  ? 

13.  A  cord  of  good  walnut  wood  may  be  obtained  for  8  dollars ; 
what  must  I  give  for  4  cords  ?     For  6  cords  ?     For  9  cords  ? 

14.  What  cost  4  quarts  of  milk  at  5  cents  a  quart,  and  8  gal- 
lons of  vinegar  at  10  cents  a  gallon  ? 

15.  If  a  man  earn  7  dollars  a  week,  how  much  will  he  earn 
in  3  weeks  ?  In  4  weeks  ?  In  5  weeks  ?  In  6  weeks .''  In  7 
weeks  ?     In  9  weeks  ? 

16.  If  1  thousand  feet  of  boards  cost  12  dollars,  what  cost  4 
thousand  ?  5  thousand  ?  6  thousand  ?  7  thoustuidi"  9  thou- 
sand ?     12  thousand? 

17.  If  3  pairs  of  shoes  buy  one  pair  of  boots,  how  many  pairs 
of  shoes  will  it  take  to  buy  7  pairs  of  boots  ? 

18.  If  5  bushels  of  apples  buy  1  barrel  of  flour,  how  many 
bushels  of  apples  are  equal  in  value  to  12  barrels  of  flour? 

19.  If  one  yard  of  canvas  cost  25  cents,  what  will  12  yards 
cost  ?  "J 

Illustration.  —  25  is  composed  of  2  tens  and  5  units ;  12  times  2 
tens  are  24  tous;  and  12  times  5  units  are  60  units,  or  6  tens,  2  4  tens 


MULTIPLICATION.  85 

added  to  6  tens  make  30  tens,  or  300.     Therefore,  12  yards  will  cost 
300  cents. 

20.  In.  1  pound  there  are  20  shillings ;  how  many  shillings  in 
3  pounds?     In  4  pounds?     In  6  pounds? 

21.  A  gallon  of  molasses  is  worth  25  cents ;  what  is  the 
value  of  2  gallons  ?     Of  4  gallons  ?     Of  9  gallons  ? 

22.  If  1  man  earn  12  dollars  in  16  days,  how  much  would 
10  men  eani  in  the  same  time  ? 

23.  If  a  steam-engine  runs  28  miles  in  1  hour,  how  far  will 
it  run  in  4  hours  ?     In  6  hours  ?     In  9  hours  ? 

24.  If  the  earth  turns  on  its  axis  15  degrees  in  1  hour,  how 
far  will  it  turn  in  7  hours?     In  11  hours?     In  12  hours? 

25.  In  a  certain  regiment  there  are  8  companies,  in  each 
company  6  platoons,  and  in  each  platoon  12  soldiers ;  how  many 
soldiers  are  there  in  the  regiment? 

2G.  If  1  man  walk  7  miles  in  1  hour,  how  far  will  he  walk 
in  8  hours?     In  11  hours?     In  20  hours?     In  30  hours? 

35.  Multiplication  is  the  process  of  taking  a  number  as  many 
times  as  there  are  units  in  another  number. 

In  multiplication  three  terms  are  employed,  called  the  MuU 
tiplicand,  the  Multiplier,  and  the  Product. 

The  Multiplicand  is  the  number  to  be  multiplied  or  taken. 

The  Multiplier  is  the  number  by  which  Ave  multiply,  and  de- 
notes the  number  of  times  the  multiplicand  is  to  be  taken. 

The  Product  is  the  result,  or  number  produced  by  the  multi- 
pliciition. 

The  multiplicand  and  multiplier  are  often  called  Factors. 

Note.  —  The  multiplicand  may  be  either  an  abstract  or  concrete  number, 
but  the  multiplier  must  always  be  regarded  as  an  abstract  numbei-.  The 
product  is  of  J;he  same  kind  as  the  multiplicand. 

The  Sign  of  Multiplication  is  formed  by  two  short'  lines  cross- 
ing each  other  obliquely ;  thus,  X«  It  shows  that  the  numbers 
between  which  it  is  placed  are  to  be  multiplied  together ;  thus, 
7  X  5  =  35  is  read,  7  multiplied  by  5  is  equal  to  35. 


35.  What  is  multiplication  ?  What  three  terms  are  employed  1  What  is 
the  multiplicand  ?  The  multiplier  ?  The  product?  What  are  the  multi- 
plicand and  multiplier  often  called  ?  The  sign  of  multiplication  1  What 
does  it  show  1 


36 


MULTIPLICATION. 


36.   When  the  multiplier  does  not  exceed  12. 

Ex.  1.  Let  it  be  required  to  multiply  175  by  7.    Ans.  1225. 


OPERATION. 


Multiplicand    17  5 
Multiplier  7 

Product        12  2  5 


Having  -written  the  multiplier  under  the 
unit  figure  of  the  multiplicand,  we  multiply 
the  5  units  by  7,  obtaining  35  units,  or  3 
tens  and  5  units,  and  set  down  the  5  units 
under  the  7,  and  reserve  the  3  tens  for  the 
tens'  column.  We  then  multiply  the  7  tens 
by  7,  obtaining  49  tens,  and,  adding  the  3  tens  Avhich  were  reserved, 
we  have  52  tens,  or  5  hundreds  and  2  tens.  Writing  down  the  2  lens, 
and  reserving  the  5  hundreds,  we  multiply  the  1  hundred  by  7  ;  and, 
adding  the  reserved  5  hundreds,  we  have  twelve  hundreds,  which  we 
write  down  in  full ;  and  the  product  is  1225. 


Examples  for  Practice. 


Multiply 
By 

Ans. 


2. 

875  6 
4 

35024 


3. 

4567 
3 

13701 


4. 

7896 
5 

3  9480 


5. 

56807 
5 

284035 


47893 
6 

287358 


7. 
61657 

7 

4  3  15  9  9 


8. 
8  9  7  6  5 
9 

807885 


9.  Multiply  767853  by  9. 

10.  Multiply  876538765  by  8. 

11.  Multiply  7654328  by  7. 

12.  INIultiply  4976387  by  5. 

13.  Multiply  8765448  by  12. 

14.  Multiply  4567839  by  11. 

15.  What  cost  8675  barrels  of  flour  at  7  dollars  per  barrel  ? 

Ans.  60725  dollars. 


Ans.  6910677. 

Ans.  7012310120. 

Ans.  53580296. 

Ans.  24881935. 

Ans.  105185376. 

Ans.  50246229. 


36.  How  must  numbers  be  written  for  iiinltiplication  ?  At  wbirli  hand  do 
you  bcpin  to  multiply  /  Why  ?  Where  do  yi>n  write  the  first  or  riizht-hand 
ii;^Mire  (if  llie  jirodiict  of  ea<-h  liLTure  in  ilie  iiitihi])iieaiid  '.  Why  ?  What  is 
done  with  the  tens  or  Itfl-hand  ti;;nre  of  eaeh  produet  i  How,  then,  do  you 
proceed  when  tlic  multiplier  does  not  exceed  12  ? 


MULTIPLICATION.  37 

16.  "Wliat  cost  25384  tons  of  hay  at  9  dollars  per  ton  ? 

Ane.  22845 G  dollars. 

17.  If  on  1  page  in  this  book  there  are  2538  letters,  how 
many  are  there  on  11  pages?  Ans.  27918  letters. 

37.  When  the  multiplier  exceeds  12. 

Ex.  1.   Let  it  be  required  to  multiply  763  by  24. 

Ans.  18312. 

oPEKATiox.  We   write  the   multiplier  under   the 

Multiplicand       7  6  3         multiplicand,  and   proceed   to   multiply 
Multipher  2  4        *^  multiplicand  by  4  the  unit  figure  of 

^  the  multiplier,  as  in  Ai't.  36.     We  then, 

3  0  5  2         in  like  manner,  multiply  the  multiplicand 
15  2  6  by  the  2  tens  in  the  multiplier,  taking 

care  to  write  the  first  figure  obtained  by 

Product         18  3  12         this  multiplication  in  tens'  column,  di- 
rectly  under  the   2  of  the   multiplier ; 
and,  adding  the  partial  products  obtained  by  the  two  multiplications, 
we  find  the  whole  product  of  7(J3  multiplied  by  24  to  be  18312. 

38.  Rule.  —  Write  the  multiplier  under  the  multiplicand,  arranging 
units  under  units,  tens  under  tens,  S)'c. 

If  the  mulliplier  is  one  figure,  multiply  each  figure  of  the  multiplicand 
in  succession,  beginning  with  the  units'  figure,  by  the  multiplier,  writing 
the  right-hand  figure  of  each  product  under  the  figure  multiplied,  and 
adding  the  left-hand  figure,  if  any,  to  the  succeeding  product;  but 
observing  to  ivrite  down  all  the  figures  of  the  last  product. 

If  the  midtiplier  contains  more  than  one  figure,  multiply  by  each  figure 
separately,  loriting  its  product  in  a  separate  line,  and  observing  to  place 
the  right-hand  figure  of  each  line  under  the  figure  by  which  you  multiply. 

Tlie  sum  of  the  several  products  will  be  the  ivhole  product  reqiured. 

Note.  —  "When  there  are  ciphers  between  the  sij^nificant  figures  of  the 
multiplier,  pass  over  them  iu  the  operation,  and  multiply  by  the  significant 
figures  only. 


37.  How  flo  you  proceed  when  the  mixltiplior  exceeds  12  ?  "Where  do  you 
set  tlic  first  fieiire  of  each  partial  product  ?  Why  ?  How  is  the  true  product 
found  1  — .38.  The  general  rule  for  multiplication  1  When  there  are  ciphers 
between  the  significant  figures  of  the  multiplier,  how  do  you  proceed  1 

4 


38  MULTIPLICATION. 

39.  First  Method  of  Proof.  —  Multiply  the  multiplier  by  the 
multiplicand,  and  if  the  result  is  like  the  lirsst  product,  the  work  is 
supposed  to  be  right. 

Note.  —  This  proof  depends  on  the  principle,  That,  when  two  or,  more 
numbers  are  multiplied  together,  the  product  is  the  same,  whatever  the  order  of 
multiplying  them. 

Ex.  2.     Multiply  7895  by  56. 

OPERATION. 

Multiplicand         7  8  9  5 
Multiplier  5  6 


47370 
39475 


Product  4  4  212  0 


Ans.  442120. 

PROOF. 

56 

7895 

280 

504 

448 

392 

Product        4  4  2  12  0 


Note.  —  The  common  mode  of  proof  in  business  is  to  divide  the  product 
bv  the  multiplier,  and,  if  tiie  work  is  right,  the  quotient  will  be  like  the  mul- 
tiplicand. This  mode  of  proof  anticipates  the  principles  of  division,  and 
therefore  cannot  be  employed  without  a  previous  knowledge  of  that  rule. 

40.  Second  Method  of  Proof  —  Beginning  at  the  left  hand  of 
the  multiplicand,  add  together  its  successive  figures  towards  the 
right  till  the  sum  obtained  equals  or  exceeds  nine.  Omit  the 
nine,  and  carry  the  excess,  if  any,  to  the  next  figure.  Proceed 
in  this  way  till  all  the  figures  in  the  multiplicand  have  been  added, 
and  write  the  final  excess  at  the  right  hand  of  the  multiplicand. 

Proceed  in  a  like  manner  with  the  multiplier,  and  write 
the  final  excess  uader  that  of  the  multiplicand.  Multiply  tlie>e 
excesses  together,  and  place  the  excess  of  nines  in  their  product 
at  the  right. 

Find  the  excess  of  nines  in  the  product  obtained  by  the  origi- 
nal operation  ;  and,  if  the  work  is  right,  the  excess  thus  found 


.39  TIow  is  multiplication  proved  bv  the  first  method  ?  What  i<  the  rea- 
son ffir  this  method  ''  Wliat  is  tlio  ronimr)ii  mode  of  proof  in  busiuess  I  — ' 
40.  What  is  tlic  second  method  of  proving  multiplication  ? 


MULTIPLICATION.  _  39 

Will  be  equal  to  the  excess  contained  in  the  product  of  the  ex- 
cesses of  the  multiplicand  and  multiplier. 

Ex.3. 

Multiplicand         12  3  4  5=  6  excess. 

MultipUer  2  2  3  1=  8  excess. 

12345      48  =  3 

37035 

24690  I   T,      - 

2  4  6  9  0  ^  ^^°<'^- 


Product       27541695=  3 

Note.  —  This  method  of  proof,  thoufjh  perhaps  sufficiently  sure  for  com- 
mon purposes,  is  not  always  a  test  of  the  correctness  of  an  operation.  If 
two  or  more  figures  in  the  work  should  be  transposed,  or  the  value  of  one 
figure  be  just  as  much  too  great  as  another  is  too  small,  or  if  a  nine  be  set 
down  in  the  place  of  a  cipher,  or  the  contrary,  the  excess  of  nines  will  be 
the  same,  and  still  the  work  may  not  be  correct.  Such  a  balance  of  errors 
will  not,  however,  be  likely  to  occur. 

Examples  for  Practice. 

4.  5. 

Multiply      67895  78956 

By  3  6  4  7 

407370  552692 

203685  315824 


Ans.   2444220  3710932 

6.  7. 

Multiply    893  25  47896 

By          901  2008 

89325  383168 

8039  2  5  95792 


Ans.   80481825  96175168 

8.  "What  cost   47   hogsheads  of  molasses  at  13  dollars   per 
hogshead?  Ans.  611  dollars. 

9.  What  cost  97  oxen  at  29  dollars  each  ?   Ans.  2813  dollars. 


40.  Is  this  method  of  proof  always  a  true  test  of  the  correctness  of  an 
operation  1     The  reason  for  this  method  of  proof? 


40  MULTIPLICATION. 

10.  Sold  Jt  farm  containing  367  acres,  at  97  dollars  j)er  acre; 
what  was  the  amount  ?  Ans.  35599  dollars. 

11.  An  army  of  17006  men  receive  each  109  dollars  as  their 
annual  pay ;  what  is  the  amount  paid  the  whole  army  ? 

Ans.  1853654  dollars. 

12.  If  a  mechanic  deposit  annually  in  the  Savings  Bank  407 
dollars,  what  will  be  the  sum  deposited  in  27  years  ? 

Ans.  10989  dollars. 

13.  If  a  man  travel  37  miles  in  1  day,  how  far  will  he  travel 
in  365  days  ?  Ans.  13505  miles. 

14.  If  there  be  24  hours  in  1  day,  how  many  hours  in  365 
days?  Ans.  8760  hours. 

15.  How  many  gallons  in  87  hogsheads,  there  being  63  gal- 
lons in  each  ?  Ans.  5481  gallons. 

16.  If  the  expenses  of  the  Massachusetts  Legislature  be  1839 
dollars  per  day,  what  will  be  the  amount  in  a  session  of 
109  days?  Ans.  200451  dollars. 

17.  If  a  hogshead  of  sugar  contains  368  pounds,  how 
many  pounds  in   187  hogsheads?  Ans.  68816  pounds. 

18.  Multiply  675  by  476.  Ans.  321300. 

19.  Multiply  679  by  763.  Ans.  518077. 

20.  Multiply  899  by  981.  Ans.  881919. 

21.  Multiply  7854  by  1234.  Ans.  9691836. 

22.  Multiply  3001  by  6071.  Ans.  18219071. 

23.  Multiply  7117  by  9876.  Ans.  70287492. 

24.  Multiply  376546  by  407091.  Ans.  153288487686. 

25.  Multiply  7001009  by  7007867.   Ans.  49062139937803. 

26.  Multiply  five  hundi-ed  and  eighty-six  by  nine  hundred 
and  eight.  Ans.  532088. 

27.  Multiply  three  thousand  eight  hundred  and  five  by 
one  thousand  and  seven.  Ans.  3831635. 

28.  Multiply  two  thousand  and  seventy-one  by  seven  hun- 
dred and  six.  Ans.  1462126. 

29.  Multiply  eighty-eight  thousand  and  eight  by  three 
thousand  and   seven.  Ans.  264640056. 

30.  Multiply  ninety  thousand  eight  hundred  and  seven  by 
one  thousand  and  ninety-one.  Ans.  99070437. 

31.  Multiply  ninety  thousand  eight  hundred  and  seven  by 
nine  thousand  one  hundred  and  six.  Ans.  826888542. 

32.  ?''ultiply  fifty  thousand  and  one  by  five  thousand  eight 
hundred  and  seven.  Ans.  290355807'. 

33.  Multiply  eighty  thousand  and  nine  by  nine  thousand 
and  sixteen.  Ans.  721361144. 


MULTIPLICATION.  41 

34.  Multiply  forty-seven  thousand  and  thirteen  by  eijihty 
thousand  eight  hundred  and  seven.  Ans.  371)897941)1. 

41  •  A  Composite  number  is  one  produced  by  multiplying 
together  two  or  more  whole  numbers  greater  than  unity  or  one  ; 
thus,  12  is  a  composite  number,  since  it  is  the  product  of  3  X  4  ; 
and  24  is  a  composite  number,  since  it  is  the  product  of  2  X  3 
X  4. 

A  Factor  of  any  number  is  a  name  given  to  one  of  two  or  more 
wliole  numbers  greater  than  unity,  which,  being  multiphed 
toge;her,  produce  that  number;  thus,  3  and  4  are  iactors  of  12, 
since  3  X  4  =  12. 

42.    To  multiply  by  a  composite  number. 

Ex.  1.  Bought  15  pieces  of  broadcloth,  at  96  dollars  per  piece; 
how  much  did  I  pay  for  the  whole  ?  Ans.  1440  doUai's. 

OPERATION.  The  factors  of  15  are  3  and 

9  6  dolls.,  price  of  1  piece.        5.     Now,    if  we    multiply   the 

3  price  of  one  piece  by  the  factor 

3,  we  get  the  price  of  3  pieces  ; 

fv  o  o   1  n  •        CO'  and  then,   by    multiplyiii<i   the 

2  8  8  dolls.,  price  of    3  pieces.     ^^.^^  ^^  3  ^-^^.^^  by  thefoctor 

^  5,   we  obtain  the   price    of   15 

pieces,  the  number  bought,  since 

14  4  0  dolls.,  price  of  15  pieces.      15  is  equal  to  five  times  3. 

RuLK.  —  Multiphj  the  muliiplicand  hy  one  ^f  the  factors  of  the  mid- 
iipUer,  and  the  product  thus  obtained  hy  another,  and  so  on  until  each 
of  the  factors  has  been  used  as  a  multiplier.  The  last  product  will 
he  the  answer. 

Note  —  The  product  of  any  number  of  factors  is  the  same  in  whatever 
order  they  are  muUiplied.     Thus,  3X4  =  12,  and  4  X  3  =  12. 

Examples  for  Practice. 

2.  Multiply  30613  by  25  =  5  X  5.  Ans.  765325. 

3.  Multiply  1469  by  84  =  7  X  12.  Ans.  123396. 

4.  Multiply  7546  by  81,  using  its  factors.  Ans.  611226. 

5.  Multiply  7901  by  125,  using  its  factors.         Ans.  987625. 

6.  In  1  mile  there  are  63360  inches;  how  many  inches  in  45 
miles?  Ans.  2851200. 

7.  If  in  one  year  there  are  8766  hours,  how  many  hours  in 
72  years?  Ans.  631152  hours. 

41    "What  is  a  composite  number?     A  fnotor  of  any  number? — 42.  What 
are  the  factors  of  15?     How  do  we  multiply  by  a  composite  number  ?     The 
rule  ?    In  what  order  may  the  factors  of  a  composite  number  be  multiplied  J 
4* 


42  MULTIPLICATION. 

8.  If  sound  moves  1142  feet  in  a  second,  how  far  will  it  move 
in  one  minute  ?  Ans.  G8520  feet. 

43.    When  the  multipher  is  1  with  one  or  more  ciphers 
annexed,  as  10,  100,  &c. 

Ex.  1.  In  1  day  there  are  24  hours  ;  how  many  hours  in  10 
days  ?     In  100  days  ?  Answers  240,  2400. 

OPERATION.  The  removal  of  a  figure  one 

Multiplicand  2  4  2  4  place  to  the  left  makes  the  val- 

Multiplier           10          10  0  "<^  expressed  tenfold.  (Art.  7.) 

Therefore,    by    annexing    one 

Product           2  4  0  2  4  0  0  ^'Pl^^r  to  24,  the  multiplicand, 

r*      I,             ojA  oiAA  each    ngure    is    removed    one 

Ur  thus,       ^4U,  ZiOO.  ^^^^^^  ^^  ^.j^^  ^^^^^  ^^^^  ^^^  ^^^1^^ 

expressed  made  tenfold,  or  multiplied  by  10 ;  and  by  annexing  two 
ciphers,  each  figure  is  removed  two  places  to  the  left,  and  the  value 
expressed  made  one  hundred-fold,  or  multiplied  by  100. 

Rule.  —  Annex  to  the  multiplicand  as  man!/  ciphers  as  has  the  mul- 
tiplier.    The  number  thus  formed  will  be  the  product  required. 

Examples  for  Pkactice. 

2.  Multiply  2356  by  10.  "  Ans.  23560. 

3.  Multiply  5873  by  100.  Ans.  587300. 

4.  Multiply  7964  by^lOOO.  Ans.  7964000. 

5.  Multiply  98725  by  100000.  Ans.  9872500000. 

44 •  When  there  are  ciphers  on  the  right  hand  of  the 
multipher  or  multiphcand,  or  both. 

Ex.  1.  What  will  600  acres  of  land  cost  at  20  dollars  per 
acre?  Ans.  12000  dollars. 

orERATioN.  The  multiplicand  may  be  resolved  into 
INIultiplicand      6  0  0  the  factors  6  and  100,  and  the  multipher 

Multipher  2  0  '"*'°  *^^^  factors  2  and  10.     Now,  it  is  evi- 

dent  (Art.  42),  if  these  several  factors  be 

Product  12  0  0  0      "^"Itiphed  together    they  will  produce  the 

same  product  as  the  original  factors  COO 
and  20.  Thus  6  X  2  =  12,  and  12  X  100  =  1200,  and  1200  X  10  = 
12000. 

43.  Whiit  is  the  effect  of  removinp:  a  figure  one  phu>c  to  the  left?  What 
is  t!ic  effect  of  iiniicxinji  a  ciplicr"?  Two  ciplicrs  ?  &c.  The  rule  wlion  the 
imihiphcr  is  1  with  cii)iiers  aiuiexed  ?  —  44.  How  do  you  arrange  the  ilLTures 
for  iniiliiplicatioii,  when  there  are  ciphers  on  the  ri<;ht  liand  of  cither  tho 
multiplier  or  multiplicand,  or  hoth  ?  Why  docs  midtiplying  the  significant 
figures  and  annexing  tho  ciphers  produce  the  true  product  ? 


MULTIPLICATION.  43 

KuLE.  —  Write  the  sigmficant  f{jures_  of  the  multiplier  under  those  of 
the  multiplicand,  and  muUiply  them  together.  To  their  product  annex  as 
manij  ciphers  as  there  are  on  the  right  of  both  multiplicand  and  multiplier. 

Examples  for  Practice. 

2.  3. 

IMultiply    8785324  713378900 

By  3200  70080 

17570  648  57070312 

26355972         49936523 


Ans.    28113036800     49993593312000 

4.  Multiply  8010700  by  9000909.    Ans.  72103581726300. 

5.  Multiply  700110000  by  700110000. 

Ans.  490154012100000000. 

6.  Multiply  4070607  by  7007000.    Ans.  28522743249000. 

7.  Multiply  4110000  by  1017010.      Ans.  4179911100000. 

8.  Multiply  twenty-nine  millions  two  thousand  nine  hundred 
and  nine  by  four  hundred  and  four  thousand. 

Ans.  11717175236000. 

9.  Multiply  eighty-seven  milUons  by  eight  hundred  thousand 
seven  hundred.  Ans.  69660900000000. 

10.  Multiply  one  million  one  thousand  one  hundred  by  nine 
hundred  nine  thousand   and  ninety.         Ans.  910089999000. 

11.  Multiply  forty-nine  milhons  and  forty-nine  by  four  hun- 
dred and  ninety  thousand.  Ans.  24010024010000. 

12.  Multiply  two  hundred  millions  two  hundred  by  two  mil- 
lions two  thousand  and  two.  Ans.  400400800400400. 

13.  Multiply  four  millions  forty  thousand  four  hundred  by  three 
hundred  three  thousand.  Ans.  1224241200000. 

14.  Multiply  three  hundred   thousand   thirty  by  forty-seven 
thousand  seventy.  Ans.  14122412100. 

15.  Multiply  fifteen  millions  one  hundred  by  two  thousand  two 
hundred.  Ans.  33000220000. 

16.  Multiply  one  bilhon  twenty  thousand  by  one  thousand  one 
hundred.  Ans.  1100022000000. 


44.  What  is  the  rule  1 


44 


DIVISION. 


DIVISION. 

45.  When  it  Is  required  to  find  how  many  times  one  number 
contains  another,  the  process  is  called  Division. 

Ex.  1.  A  boy  has  32  cents,  which  he  wishes  to  give  to  8  of 
his  companions,  to  each  an  equal  number ;  how  many  must  each 
receive  ? 

Illustration.  —  Each  must  receive  as  many  cents  as  8,  the  num- 
ber of  companions,  is  contained  times  in  32,  the  number  of  cents.  A\'e 
therefore  inquire  what  number  8  must  be  multiplied  by  to  make  32. 
By  trial,  we  find  that  4  is  the  number  ;  because  4  times  8  are  32. 
Hence  8  is  contained  in  32  4  times,  and  each  of  his  companions  must 
receive  4  cents. 

DIVISION    TABLE. 


2    in 

2 

1  time 

3 

in 

3 

1  time 

4 

in 

4 

1  time 

5 

in 

5 

1  time 

2    " 

4 

2  times 

3 

4( 

6 

2  times 

4 

u 

8 

2  times 

5 

(( 

10 

2  times 

2    " 

6 

3    " 

3 

(t 

9 

3    " 

4 

(C 

12 

3    " 

5 

(C 

15 

3    " 

2    " 

8 

4    " 

3 

ii 

12 

4    " 

4 

u 

16 

4    " 

5 

u 

20 

4    " 

2    " 

10 

5    " 

3 

U 

15 

5    " 

4 

"- 

20 

5    " 

6 

n 

25 

5    " 

2    " 

12 

6    » 

3 

a 

18 

6    " 

4 

u 

24 

6    " 

5 

11 

30 

6    " 

2    " 

14 

7    " 

3 

(( ■ 

21 

7    " 

4 

u 

28 

7    " 

5 

(( 

35 

7    "   . 

2    " 

16 

8    " 

3 

t( 

24 

8    " 

4 

(I 

32 

8    " 

5 

11 

40 

8   " 

2    " 

18 

9    " 

3 

a 

27 

9    " 

4 

t( 

36 

9    " 

5 

11 

45 

9    " 

2    " 

20 

10    " 

3 

u 

30 

10    " 

4 

u 

40 

10    " 

5 

11 

50 

10  "• 

2    " 

22 

11    " 

3 

a 

33 

11    " 

4 

u 

44 

11    " 

5 

11 

55 

11    " 

2    " 

24 

12    " 

3 

it 

36 

12    " 

4 

a 

4S 

12    " 

5 

11 

60 

12   " 

6    in 

6 

1  time 

7 

in 

7 

1  time 

8 

in 

8 

1  time 

9 

in 

9 

1  time 

6    " 

12 

2  times 

7 

l; 

14 

2  times 

8 

ti 

16 

2  times 

9 

11 

18 

2  times 

6    " 

18 

3    " 

7 

(( 

21 

3    " 

8 

a 

24 

3    •' 

9 

11 

27 

3    " 

6    " 

24 

4    " 

7 

ik 

28 

4    " 

8 

ti 

32 

4    " 

9 

It 

36 

4    " 

6    " 

30 

5    " 

7 

u 

35 

6    " 

8 

u 

40 

5    " 

9 

11 

45 

5    " 

6    " 

3'! 

6    " 

7 

(t 

42 

6    " 

8 

a 

48 

6    " 

9 

11 

64 

6    " 

6    " 

42 

7    " 

7 

i( 

49 

7    " 

8 

(I 

56 

7    " 

9 

11 

63 

7    " 

6    " 

48 

8    " 

7 

l( 

56 

8    " 

8 

(i 

64 

8    " 

9 

11 

72 

8   " 

6    " 

54 

9    " 

7 

l( 

03 

9    " 

8 

a 

72 

9    " 

9 

11 

81 

9   " 

6    " 

GO 

10    " 

7 

n 

70 

10    " 

8 

a 

80 

10    " 

9 

11 

90 

10   " 

6    " 

66 

11    " 

7 

(C 

77 

11    " 

8 

a 

88 

11    " 

9 

(1 

99 

n  " 

6    " 

72 

12   " 

7 

n. 

84 

12    " 

8 

u 

98 

12    " 

9 

11 

108 

12   '< 

10    in 

10 

1  time 

10 

in 

110 

11  times 

11 

in 

77 

7  times 

12 

in 

36 

S  times 

10    " 

20 

2  times 

10 

u 

120 

12    " 

11 

a 

88 

8   " 

12 

*' 

48 

4    " 

10    " 
10    " 

30 
40 

3  " 

4  " 

11 
11 

99 
110 

9    " 
10    " 

12 
12 

11 
11 

60 
72 

5  " 

6  " 

10    " 

50 

5    " 

n 

in 

11 

1  time 

11 

ti 

121 

11    " 

12 

11 

84 

7    " 

10    " 

60 

6    " 

n 

u 

22 

2  times 

11 

i( 

132 

12    " 

12 

11 

96 

8    " 

10    " 

10     " 

70 
80 

7  " 

8  " 

n 
11 

(( 

li 

33 
44 

3  " 

4  •' 

12 
12 

11 

108 
120 

9    " 
10    " 

10    " 

!tO 

9    " 

11 

i( 

55 

5    " 

12 

in 

12 

1  time 

12 

11 

132 

n  " 

10    " 

100 

10    " 

11 

u 

66 

6    " 

12 

it 

24 

2  times 

12 

11 

144 

12    " 

DIVISION.  45 

2.  A  farmer  received  8  dollars  for  2  sheep ;  what  was  the 
price  of  each? 

Illustration.  —  Since  he  received  8  dollars  for  2  sheep,  for  1  sheep 
he  must  receive  as  many  dollars  as  2  is  contained  times  in  8.  2  is  con- 
tained in  8  4  times,  because  4  times  2  are  8  ;  hence  4  dollars  was  the 
price  of  each  sheep. 

3.  A  man  gave  15  dollars  for  3  barrels  of  flour;  what  was  the 
cost  of  each  barrel  ? 

4.  A  lady  divided  20  oranges  among  her  5  daughters  ;  how 
many  did  each  receive  ? 

5.  If  4  casks  of  lime  cost  12  dollars,  what  costs  1  cask  ? 

6.  A  laborer  earned  48  shillings  in  6  days  ;  what  did  he  re- 
ceive per  day  ?  ^ 

7.  A  man  can  perform  a  certain  piece  of  labor  in  30  days ; 
how  long  will  it  take  five  men  to  do  the  same  ? 

8.  When  72  dollars  are  paid  for  8  acres  of  land,  what  costs  1 
acre  ?     What  cost  3  acres  ? 

9.  K  21  pounds  of  flour  can  be  obtained  for  3  dollars,  how 
much  can  be  obtained  for  1  dollar  ?  How  much  for  8  dollars  ? 
How  much  for  9  dollars  ? 

10.  Gave  56  cents  for^  8  pounds  of  raisins ;  what  costs  1 
pound  ?     What  cost  7  pounds  ? 

11.  If  a  man  walk  24  miles  in  6  hours,  how  far  will  he  walk 
in  1  hour  ?     How  far  in  10  hours  ? 

12.  Paid  56  dollars  for  7  hundred  weight  of  sugar;  what  costs 
1  hundred  weight  ?     What  cost  10  hundred  weight  ? 

13.  If  5  horses  will  eat  a  load  of  hay  in  1  week,  how  long 
would  it  last  1  horse  ? 

14.  In  20,  how  many  times  2?  How  many  times  4?  How 
many  times  5  ?     How  many  times  10  ? 

15.  In  24,  how  many  times  3  ?  How  many  times  4:?  How 
many  times  6  ?     How  many  times  8  ? 

16.  How  many  times  7  in  21  ?  In  28  ?  In  56  ?  In  35  ?  In 
14?     In  63?     In  77?     In  70  ?     In  84? 

17.  How  many  times  6  in  12?  In  36?  In  18?  In  54? 
In  60?    In  42?     In  48?.    In  72?     In  66? 

18.  How  many  tunes  9  in  27  ?  In  45  ?  In  63  ?  In  81  ? 
In  99?     In  108? 

19.  How  many  times  11  in  22?  In  55?  In  77?  In  88? 
In  110?     In  132? 

20.  How  many  times  12  in  36?  In  60  ?  In  72  ?  In  84? 
In  120?     In  144? 


46  DIVISION. 

46t  Division  is  the  process  of  finding  how  many  times  one 
number  is  contained  in  another ;  or  the  process  of  separating  a 
number  into  a  proposed  number  of  equal  parts. 

In  division  there  are  three  principal  terms  :  the  Dividend,  the 
Divisor,  and  the  Quotient,  or  Answer. 

The  Dividend  is  the  number  to  be  divided. 

The  Divisor  is  the  number  by  which  we  divide. 

The  (Quotient  is  the  number  of  times  the  divisor  is  contained  in 
the  dividend ;  or  one  of  the  equal  parts  into  which  it  is  divided. 

When  the  dividend  does  not  contain  the  divisor  an  exact  num- 
ber of  times,  the  excess  is  called  a  Remainder,  and  may  be  re- 
garded as  a  fourth  term  in  the  division. 

The  remainder,  being  part  of  the  dividend,  will  always  be  of 
the  same  denomination  or  kind  as  the  dividend,  and  must  always 
be  less  than  the  divisor. 

Note.  —  "Wlien  the  divisor  and  dividend  are  of  the  same  kind,  the  quotient 
will  be  an  abstract  number';  and  when  they  are  not  of  the  same  kind,  the 
quotient  will  be  of  the  same  kind  as  the  dividend. 

47i  The  Sign  of  Division  is  a  short  horizontal  line,  with  a  dot 
above  it  and  another  below ;  thus,  -r-.'  It  shows  that  the  number 
before  it  is  to  be  divided  by  the  number  after  it.  Thus  6^2 
=  3  is  read,  6  divided  by  2  is  equal  to  3. 

Division  is  also  indicated  by  writing  the  dividend  above  a  short 
horizontal  line  and  the  divisor  below ;  thus,  ■§  =  3  is  read,  6 
divided  by  2  is  equal  to  3. 

There  is  a  third  method  of  indicating  division,  by  a  curved 
line  placed  between  the  divisor  and  dividend.  Thus,  the  expres- 
sion 6)  12  shows  that  12  is  to  be  divided  by  6. 

48.  Sliort  Division,  or  when  the  divisor  does  not  exceed 
12. 

Ex.  1.    Divide  8574  dollars  equally  among  6  men. 

An?.  1429  dollars. 


46.  What  is  division  1  Wliat  arc  the  throe  principal  terms  in  division? 
Wliat  is  the  dividend  ?  The  divisor?  What  is  tlic  quotient  ?  The  remain- 
der 7  What  will  lie  the  denomination  of  the  remaiiidor?  How  does  it  com- 
pare with  the  divi-;or?  — 47.  Wliat  is  the  first  sijrn  of  division,  and  what  does 
it  show?  What  is  the  second,  nnd  what  docs  it  show  ?  What  is  the  third, 
and  vviiat  does  it  show  1  —  48.  What  is  short  division  1 


DIVISION.  47 

oritRATioN.  We  first  inquire  how  many  times 

Divisor  6  )  8574  Dividend.     6,  the  divisor,  is  contained  in  8,  the 

•  first  figure  of  the  dividend,  which 

1429     Quotient.     '^.    tliousands,   and  find  it  to  be   1 
^  tmie,  and    2   thousands   remannng. 

^Ye  write  the  1  directly  under  the  8,  its  dividend,  for  the  thousands' 
figure  of  the  quotient.  To  5,  the  next  figure  of  the  dividend,  which 
ishundreds,  we  regard  as  prefixed  the  2  thousands  ji^maining,  which 
equal  20  hundreds,  thus  forming  25  hundreds,  in  which  we  find  the 
divisor  6  to  be  contained  4  times,  and  1  hundred  remaining.  We 
write  the  4  for  the  hundreds'  figure  in  the  quotient,  and  the  1  hundred 
remaining,  equal  10  tens,  we  regard  as  prefixed  to  7,  the  next  figure  of 
tlie  dividend,  which  is  tens,  forming  17  tens,  in  which  the  divisor  6 
is  contained  2  times,  and  5  tens  remaining.  We  write  the  2  for  the 
tens'  figure  in  the  quotient,  and  the  5  tens  remaining,  equal  50  units, 
we  regard  as  jjrefixed  to  4,  the  last  figure  of  tlie  dividend,  which  is 
units,  forming  54  units,  in  which  the  divisor  6  is  contained  9  times. 
Writing  the  9  for  the  units'  figure  of  the  quotient,  we  have  1429  as  the 
entire  quotient. 

49t  E.ULE.  —  Write  the  divisor  at  the  left  hand  of  the  dividend, 
with  a  curved  line  between  them,  and  draw  a  horizontal  line  under  the 
dividend. 

TJicn,  beginning  at  the  left,  find  how  many  times  the  divisor  is  con- 
tained in  the  fewest  figures  of  the  dividend  that  will  contain  it,  and  write 
the  quotient  under  its  dividend. 

If  there  be  a  remainder,  regard  it  as  prefixed  to  the  next  figure  of  tlie 
dividend,  and  divide  as  before. 

Should  any  dividend  be  less  than  the  divisor,  write  a  cipher  in  the  quO" 
tient,  and  annex  another  figure,  if  any  remains,  for  a  new  dividend. 

Note  1.  —  When  there  is  a  remainder  after  dividing  the  last  figure  of  tha 
dividend,  write  it  with  the  divisor  underneath,  with  a  line  between  them,  at 
the  right  of  the  quotient. 

Note  2.  —  Prefix  means  to  place  before  ;  annex,  to  place  after. 

50i  First  Method  of  Proof .  —  Multiply  the  divisor  and  quo- 
tient together,  and  to  the  product  add  the  remainder,  if  any, 
and,  if  the  work  is  right,  the  result  obtained  will  equal  the  div- 
Mend. 


48.  How  are  the  numbers  an-anged  for  short  division  1  At  which  hand  do 
you  begin  to  divide  ?  Why  not  begin  at  the  right,  where  you  begin  to  mul- 
tiply ?  Where  do  you  write  the  quotient  ?  If  there  is  a  remainder  after 
dividing  a  figure,  what  is  done  with  it  ?  —  49.  The  rule  for  short  division? 
Repeat  the  notes?* 


48 


DIVISION. 


Note.  —  This  method  of  proof  depends  upon  the  fact,  that  division  is  the 
reverse  of  muhiplication.  The  dividend  answers  to  the  product,  the  divisor  to 
one  of  Xhc  factors,  and  the  quotient  to  the  other. 

Examples  for  Practice. 


2.  Divide  6375  by  5. 


OPERATION. 

Divisor  5)6375  Dividend. 
12  7  5  Quotient. 

PROOF. 

12  75  Quotient 
5  Divisor. 

6  3  7  5  Dividend. 

3.                               4. 

3)7893762          4)4763256 

5. 

5)3789565 

2631254               1190814 

6.                                7. 

6)876  5  389             7)98763  5 

8. 

8)378532 

9.                               10. 
9)8953784        11)7678903 

11. 
12)6345321 

12.  Divide      479956  by    6. 

13.  Divide      385678  by   7. 

14.  Divide      438789  by   8. 

15.  Divide    1678767  by   9. 

16.  Divide  11497583  by  12. 

Quotients. 

79992f 

55096^ 

548481 

1865291 

958131-|.^ 

17.  Divide  5678956  by   5. 

18.  Divide  1135791  by   7. 

19.  Divide  1622550  by   8. 

20.  Divide  2028180  by   9. 

21.  Divide  2253530  by  12. 
*22.  Divide  1877940  by  11. 

Quotients.            Rem. 
1 

6 
6 
8 
2 

9 

Sum  of  the  quotients, 

2084732         27 

.10.  TTow  i«  short  division  proved  ?  Of  what  i«  division  the  reverse  ?  To 
what  do  the  three  terms  in  division  answer  in  multiplication  ?  What,  then, 
is  the  reason  for  this  proof  of  division  1 


DIMSION.  49 

23.  Divide  944,580  dollars  equally  among  12  men,  and  what 
will  be  the  share  of  each  ?  Ans.  78,715  dollars. 

24.  Divide  154,503  acres  of  land  equally  among  9  person?. 

Ans.  17,167  acres. 

25.  A  plantation  in  Cuba  was  sold  for  7,011.608  dollars,  and 
the  amount  was  divided  among  8  persons.  What  was  paid  to 
each  person?  Ans.  876,451  dollars. 

26.  A  prize  valued  at  178,656  dollars  is  to  be  equally  divided 
among  12  men ;  what  will  be  the  share  of  each  ? 

Ans.  14,888  dollars. 

27.  Among  7  men,  67,123  bushels  of  wheat  are  to  be  dis- 
tributed; how  many  bushels  will  each  man  receive  ? 

Ans.  9,589  bushels. 

28.  If  9  square  feet  make  1  square  yard,  how  many  yards  in 
895.317  square  feet?  Ans.  99,483  yards. 

29.  A  township  of  876,136  acres  is  to  be  divided  among  8  per- 
sons ;  how  many  acres  will  be  the  portion  of  each  ? 

Ans.  109,517  acres. 

30.  Bought  a  farm  for  5670  dollars,  and  sold  it  for  7896  dol- 
lars, and  I  divide  the  net  gain  among  6  persons  ;  what  does  each 
receive?  Ans.  371  dollars. 

31.  If  6  shillings  make  a  dollar,  how  many  dollars  in  7890 
shillings?  Ans.  1315  dollars. 

51.  Long  Division,  or,  in  general,  when  the  divisor  ex- 
ceeds 12. 

Ex.  1.  A  gentleman  divided  896  dollars  equally  among  his 
7  children  ;  how  much  did  each  receive?        Ans.  128  dollars. 

OPERATION.                                •  Having  set  down  the  divisor 

Dividend  ^"^  dividend  as  in  short  divi- 

D'7\o(\c'/^nor\     i'     *.  ^ion,  we  draw  a  curved  line  at 

ivisor  7)896(128  Quotient.  ^^^  ^ight  of  the  dividend,  to 

mark  the   pla»e  for  the  quo- 

2  g  tient.     We  then  inquire  how 

,    .  many  times  7,  the  divisor,  is 

^  ^  contained  in  8,  the  first  figure 

Kg  of  the  dividend,  which  is  liun- 

dretfs ;  and,  finding  it  to  be  1 
"  "  hundred  times,  we  write  the  1 

for  the  hundreds'  fijure  in  the 


."Jl    What  is  lonfr  division?     The  difForence  bcnveen  lonjr  divi-;ion  and 
shoit  divi-;ion  1     How  do  yon  arrantrc  tlie  niunlicrs  for  long:  division  1     What 
do  you  fii-st  do  afrer  arranfrinsx  the  numbers  for  long  diviaion  ?     Where  do 
you  place  the  figures  of  the  quotient  1 
5 


60  DIVISION. 

quotient,  and  multiply  the  divisor,  7,  by  it,  writing  the  product,  7,  under 
the  8,  from  which  we  subtract  it,  and  to  the  remainder,  1,  annex  the 
9  of  the  dividend,  making  19  tens.  We  now  inquire  how  many  times 
7  is  contained  in  the  19,  and  write  the  2  obtained  for  the  tens'  figure 
of  the  quotient.  We  then  multiply  the  divisor  by  it,  and  place  the 
product  under  the  19,  and  subtract  as  before  ;  and  to  the  remainder, 
6,  we  annex  the  6  of  the  dividend,  making  56  units.  We  proceed 
next  to  find  the  units'  figure,  and,  after  subtracting  the  product  of  the 
divisor  multiplied  by  it  from  56,  find  there  is  no  remainder.  Hence 
each  one  of  the  7  children  must  receive  1 28  dollars. 

Note.  —  The  preceding  example  and  the  four  that  follow  are  usually 
performed  by  short  division,  but  are  here  introduced  to  illustrate  more  clearly 
the  method  of  operation  by  long  division. 

EXA3IPLES    FOR   PRACTICE. 

2.  Divide  1728  by  8.  Ans.  216. 

3.  Divide  987656  by  11.  Ans.  89786ia. 

4.  Divide  123456789  by  9.  Ans.  13717421. 

5.  Divide  390413609  by  12.  Ans  32534467^2. 

Ex.  6.     A  gentleman  divided  4712  dollars  equally  among  Ins 
19  sons  ;  what  was  the  shai'e  of  each  ?  Ans.  248  dollars. 

OPERATION.  We   first    inquire   how 

Dividend.  many  times  1 9,  the  divisor, 

Divisor  19)4712(248  Quotient,      is  contained  in  47,  the  two 

go  left-hand  figures  of  the  div- 

idend ;  and,  finding  it  to  be 

9  1  2  times,  we  write  the  2  in 

7  6  the  quotient,  multiply  the 

divisor  by  it,  and  subtract 

15  2  the  product  from  the  4  7  ; 

15  2  and  to  the  remainder,  9, 

■  annex  1,  the  next  figure  of 

the  dividend,  making  91.  We  next  inquire  how  many  times  19  is  con- 
tained in  91,  place  the  number,  4,  in  the  quotient,  then  multiply  and 
subtract  as  before,  and  to  the  remainder,  15,  annex  2,  the  last  figure 
of  the  dividend,  and,  proceeding  in  like  manner  as  before,  after  finding 
the  quotient  figure,  there  is  no  remainder.  Hence  the  share  of  each 
of  the  19  sons  is  248  dollars.  This  illustration,  except  in  omissions,  is 
essentially  like  the  preceding  one. 

."51.  After  the  quotiont  fiRure  is  fonml,  what  is  the  next  thincr  you  do? 
Where  do  you  place  the  product  ?  What  do  you  next  do  ?  What  is  the 
next  step  ?  '  How  do  you  then  proceed  1  Is  long  division  the  same  in  prin- 
ciple as  short  division  ? 


DIVISION.  51 

52*  Rule,  -*-  Write  the  divisor  and  dividend  as  in  short  division, 
and  draw  a  curved  line  at  the  right  hand  of  the  dividend. 

Then  inquire  how  many  times  the  divisor  is  contained  in  the  fewest 
figures  on  the  left  hand  of  the  dividend  that  will  contain  it,  and  write  the 
result  at  the  right  hand  of  the  dividend  for  the  first  quotient  figure, 

Muliiphj  the  divisor  hij  the  quotient  figure,  and  subtract  the  product 
from  the  figures  of  the  dividend  used,  and  to  the  remainder  annex  the  next 
figure  of  the  dividend. 

Find  how  many  times  the  divisor  is  contained  in  the  number  thus 
formed ;  ivrite  the  figure  denoting  it  at  the  right  hand  of  the  former 
quotient  figure. 

Thus  proceed  until  all  the  figures  of  the  dividend  are  divided. 

Note  1 .  —  If,  when  a  figure  is  brought  down,  the  number  formed  will  not 
contain  the  divisor,  a  cipher  must  be  placed  in  the  quotient,  and  another 
figure  of  the  dividend  brought  down,  and  so  on  until  the  ntunber  is  large 
enough  to  contain  the  divisor. 


•&' 


Note  2.  —  If  there  is  a  remainder  after  dividing  all  the  figures  of  the 
dividend,  it  must  be  written  as  directed  in  the  preceding  rule.  (Art.  49, 
Note  1.) 

NoTK  3.  —  The  proper  remainder  is  in  all  cases  less  than  the  divisor.  If, 
in  the  course  of  the  operation,  it  is  at  any  time  found  to  be  as  large  as,  or 
larger  than,  the  divisor,  it  will  show  that  there  is  an  error  in  the  work,  and 
that  the  quotient  figure  should  be  increased. 

Note  4.  —  If,  at  any  time,  the  divisor,  multiplied  by  the  quotient  figure, 
produces  a  product  larger  than  the  part  of  the  dividend  used,  it  shows  that 
the  quotient  figure  is  too  large,  and  must  be  diminished. 

53.  Second  Method  of  Proof.  —  Add  together  the  remainder, 
if  any,  and  all  the  products  that  have  been  produced  by  mukiply- 
ing  the  divisor  by  the  several  quotient  figures,  and  the  result  will 
be  like  the  dividend,  if  the  work  is  right. 

34.  TJiird  3Iethod.  —  Subtract  the  remainder,  if  any,  from  the 
dividend,  and  divide  the  difference  by  the  quotient.  The  result 
will  be  like  the  original  divisor,  if  the  work  is  right. 

Note.  —  The  first  method  of  proof  (Art.  50)  is  usually  most  conven- 
ient. 


52.  The  rule  for  lonsx  division  ?  ITnw  may  you  know  when  the  quotient 
fiirurc  is  too  small  ?  How  may  yon  know  when  it  is  too  large  ? —  53.  What 
is  the  second  mnthod  of  proof?—  54.  What  is  the  third  method?  Can  long 
division  be  proved  by  the  first  method  of  proof  (Art.  50)  1 


52 


DIVISION. 


Examples  for  Practice. 
Ex.  7.   How  many  times  is  48  contained  in  28618? 


OPERATION. 

Dividend. 
Divisor  48)28618(590  Quotient. 
240 


461 
432 


298 
288 

1  0  Remainder. 


Ans.  596. 

PEOOF    BY    JIULTIPLICATIOIt 

5  9  6  Quotient. 
4  8  Divisor. 


4768 
2384 


28608 

1  0  Remainder. 


2  8  6  18  Dividend. 


8. 

OPERATION. 

Dividend. 
Divisor  26)5  698(219  Quotient. 

*4-5  2 


49 
+2  6 


238 

+2  a  4 


PROOF   BY   ADDITION. 

52        ) 
2  6     )-  Products. 
234) 

4     Remainder. 

5  6  9  8     Dividend. 


-[-4  Remainder. 
9. 

OPERATION.  PROOF   BY   DI^^SION. 

Dividend.  Dividend. 

DiTisor  144)13824(96  Quotien*.  96)13824(144  ditUor 

1296  96 


864 
864 


422 

384 


384 
384 


10.  Divide  3276  by  14. 

11.  Divide  6205  by  17. 


Quotients. 

234 
365 


Rem. 


*  This  sign  of  addition  denotes  the  seyeral  products  to  be  added. 


DIVISION. 


53 


Quotients.  Bern. 

12.  Divide  3051  by  21.  145  6 

13.  Divide  190850  by  25.  7634  0 

14.  Divide  218570  by  42.  5204  11 

15.  Divide  9012345  by  31.  290720  25 

16.  Divide  6717890  by  98.  68549  88 

17.  Divide  4567890  by  19.  240415  5 
Divide  1357901  by  87.  15608  5 
Divide  9988891  by  77.  '  129725  66 
Divide  9999999  by  69.  144927  36 
Divide  867532  by  59.  14703  65 
Divide  167008  by  87.  1919  55 
Divide  345678  by  379.  912  30 

24.  Divide  3456789567  by  987.  3502319                 714 

25.  Divide  8997744444  by  345.  26080418  234 
Divide  4500700701  by  407.  11058232  277 
Divide  6789563  by  1234.  95 
Divide  78112345  by  8007.  4060 
Divide  34533669  by  9999.  7122 
Divide  99999999  by  3333.  0 
Divide  47856712  by  1789.  ^  962 
Divide  345678901765  by  4007.  86268755  480 
Divide  478656785178  by  56789.  8428688             22346 

34.  Divide  678957000107  by  10789561.    62927         2295060 

35.  Divide  990070171009  by  900700601.    1099     200210510 

36.  Divide  three  hundred  twenty -one  tliousand  three  hundred 
dollars  equally  among  six  hundred  seventy-five  men.    Ans.  476. 

37.  Four  hundred  seventy-one  men  purchase  a  township  con- 
taining one  hundred  eighty-six  thousand  forty-five  acres;  what 
is  the  share  of  each  ?  Ans.  395  acres. 

38.  ^  railroad,  which  cost  five  hundred  eighteen  thousand 
seventy-seven  dollars,  is  divided  into  six  hundred  seventy-nine 
shares  ;  what  is  the  value  of  each  share  ?         Ans.  763  dollars. 

39.  Divide  forty-two  thousand  four  hundred  thirty-five  bushels 
of  wheat  equally  among  one  hundred  twenty-three  men. 

Ans.,  345  bushels  each. 

40.  A  prize,  valued  at  one  hundred  eighty-four  thousand  seven 
hundred  seventy-five  dollar?,  is  to  be  divided  equally  amonsr  four 
hundred  seventy-five  men  ;  what  is  the  share  of  each?    Ans.  389. 

41.  A  certain  company  purchased  a  valuable  township  for 
nine  millions  six  hundred  ninety-one  thousand   eight  hundred 

5* 


18. 
19. 
20. 
21. 
22. 
23. 


26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 


54  DIVISION. 

thirty-six  dollars ;  each  share  was  valued  at  seven  thousand 
eight  hundred  fifty-lour  dollars ;  of  how  many  men  did  the 
company  consist  ?  Ans.  1234  men. 

42.  A  tax  of  thirty  miUions  fifty  six  thousand  four  hundred 
sixty-five  dollars  is  assessed  equally  on  four  ihousand  five  hun- 
dred ninety-seven  towns  ;  what  sum  must  each  town  pay  ? 

Ans.  C538if ^f  dollars. 

55,  Wlien  the  divisor  is  a  composito  number. 

Ex.  1.  A  merchant  bought  15  pieces  of  broadcloth  for  1440 
dollars  ;  what  was  the  cost  of  each  piece  .-*         Ans.  96  dollai-s. 

OPERATION.  The   factors   of  15    are  3 

3  )  14  4  0  dolls.,  cost  of  15  pieces,      and   5.     Now,   if  we   divide 

the  1440  dollars,  the  cost  of 

5)480  dolls.,  cost  of  5  pieces.        ^^  P'f^.f '  ^y  ,^'  ^^  obtain 
'  ^  480    dollars,    the    cost    of    5 

n  /.   1  n  .     />  1     '  pieces,   because  there  are  5 

9  6  dolls.,  cost  of  1  piece.  (jj^gs  3  in  15.     Then,  divid- 

ing 480  dollars,  the  cost  of  5  pieces,  by  5,  we  get  the  cost  of  1  piece. 

Rule.  —  Divide  the  dividend  hy  o«e  of  the  factors,  and  the  quotient 
thus  found  hy  another,  and  thus  proceed  till  every  factor  has  been  made 
a  divisor.     The  last  quotient  will  be  the  ti-ue  quotient  required. 

Examples  for  Practice. 

Quotients. 

2.  Divide  765325  by  25  =  5  X  5  30613 

3.  Divide  123396  by  84  =  7  X  12.  1469 

4.  Divide  611226  by  81,  using  its  factors.  7546 

5.  Divide  987625  by  125,  using  its  factors.  7901 

6.  Divide  17472  by  96,  using  its  factors.  182 

7.  Divide  34848  by  132,  using  its  factors.  «      264 

56.  To  find  the  true  remainder  when  there  are  several 
remainders  in  the  operation. 

Ex.  1.  How  many  months  of  4  weeks  each  are  there  in  298 
days,  and  how  many  days  remaining  ? 

Ans.  10  months  and  18  days. 


55.  Wlmt  arc  the  factors  of  15?  What  do  vou  pet  the  cost  of,  in  this 
exniiij)lo,,  when  you  divide  by  the  fiietor  ."3  ?  What,  wlicn  you  divide  by  5? 
Why  ?     Tiic  rule  for  dividinjj  hy  a  composite  number  1 


DIVISION.  55 

OPERATION.  Since  tliore  are  7  days  in  1  week, 

7)298  we    first    divide    the '  298    by    7, 

.    .  ^     .   -,  )  and    have    42    weeks    and    a    re- 

4  )  4  2,  4  aays     (  -^g  ^^^^^    malnder  of  4  days.      Then,   since 

10    2  weeks  j  •*   weeks   make    1    month,  we  di- 

'  vide   the   42   by   4,   and   have    10 

months  and  a  remainder  of  2  weeks.     Now,  to  find  the  true  remainder 

in  days,  we  multiply  the  2  weeks  by  7,  because  7  days  make  a  week, 

and  to  tlie  product  add  the  4  days ;  thus  2X7  =  14,  and  14  -[-4  =  18 

da}-s,  for  the  true  remainder. 

Rule.  —  Multiply  each  remainder,  except  the  first,  hy  all  the  divisors 
preceding  the  one  which  produced  it  ;  and  the  first  remainder  being  added 
to  the  sum  of  the  products,  the  amount  will  be  the  true  remainder. 

Note.  —  There  will  be  but  one  product  to  add  to  the  first  remainder  when 
there  are  only  two  divisors  and  two  remainders. 

Ex.  2.  Divide  789  by  36,  using  the  factors  2,  3,  and  6,  and 
find  the  true  remainder.  Ans.  33. 

OPERATION.  Dividing  by  2 

2)7  89                      5X3X2=30,  1st  Prod,  gives   394   twos, 

o  xoa  A  -,     w  15             1  X  2  =    2,  2d  Prod,  and    1   unit    re- 

6  )  oy  %  i,   ist  item.                          ^    ^^^  Rem  mainmg;   divid- 

6)T3T,   1,  2d   Rem.  _'  '     ing  the^i^os  by  3 

«f^    .        -r.  gives    131  sixes, 

2  1,  5,  3d  Rem.  33,  true  Rem.  l^^  j  ^,^,^  _  2 

remaining;  and 
dividing  the  sixes  by  6  gives  21  thirty-sixes,  and  5  sixes  =  30  re- 
maining ;  therefore  1  -[-  2  -[-  SO,  or  33,  is  the  true  remainder. 

Examples  for  Practice. 

3.  Divide  934  by  55,  using  the  factors  5  and  11,  and  find 
tlie  true  remainder.  Ans.  54. 

4.  Divide  5348  by  48,  using  the  factors  6  and  8,  and  find  the 
true  remainder.  Ans.  20. 

5.  Divide  5873  by  84,  using  the  factors  3,  4,  and  7,  and  find 
the  true  remainder.  Ans.  77. 

6.  Divide  249237  by  1728,  using  the  factors  12,  6,  6,  and  4, 
and  find  the  true  remainder.  Ans.  405. 

57.  When  the  divisor  is  1,  with  one  or  more  ciphers  at 
tlie  right. 

Ex.  1.  Divide  356  dollars  equally  among  10  men  ;  vrhat  will 
each  man  have  ?  Ans.  ^o^tj  dollars. 

56.  "When  there  are  several  remainders,  what  is  the  rule  fur  finding  the 
tiiie  remainder  ?     The  reason  for  this  rule  1 


56  DIVISION. 

it 

OPERATION.  To  multiply  by  10  vre  annex  on« 

110)3516  cipher,  which  removes  each  figure  one 

place  to  the  left,  and  thus  mak«s  the 

Quotient  3  5,        G  Rem.  value  denoted  tenfold.      Kow,  it  we 

Ov  thus    3  5  16.  reverse  the  process,  and  cut  oti"  the 

right-hand  figure  of  the  dividend  by  a 
line,  we  remove  each  remaining  figure  one  place  to  the  riyht,  and  con- 
sequently (Jiininisk  the  value  denoted  the  same  as  dividing  by  10.  Tiie 
^gures  on  the  left  of  the  line  are  the  quotient,  and  tiie  one  on  the  right 
is  the  remainder,  which  may  be  written  over  the  divisor,  and  annexed 
to  the  quotient.     Hence  the  share  of  each  man  is  3b^^  dollars. 

Examples  for  Practice. 

Quotient.  Rem. 

2.  Divide  6-892  by  10.  689  2 

3.  Divide  4375  by  100  .  75 

4.  Divide  24815  by  1000.  815 

5.  Divide  987654321123  by  100000000.  54321123 

58.    When  the  divisor  has  one  or  more  ciphers  on  the 
right,  and  is  not  10,  100,  &c. 

Ex.  1.  If  I  divide  5832  pounds  of  bread  equally  among  600 
soldiers,  what  is  each  one's  share  ?  Ans.  9|^5  pounds. 


OPERATION. 


Tlie  divisor,  GOO,  may  be  re- 
1  1  0  0  )  5  8  I  3  2  solved  into  the  factors  G  anrl  100. 

r  \  X  Q        Q  o    1  o*.  T>^^        We  first  divide  bv  the  factor  100, 
6  )  o  8,      o  2,  1st  Kem.      ,         ...         ,r  ,.  "    r  .  ^i 

<. 1  '  by  euttmg  oft  two  ngures  at  the 

9,  4,  2d  Rem.       right,  and  get  58  for  tlie  quotient 

_  /.lAANcoioo  ^^'^  ^'~  ^'^^  ^   remainder.      We 

Or  thus,  6  I  0  0  )58  |  d  2  then  divide  the  quotient,  5S,  by 

9        4  3  2  ^^^*^  other  factor,  C,  and  obtain  9 

'  for  the  quotient  and  4  for  a  re- 

mainder. The  last  remainder,  4,  being  multiplied  by  the  divi.^or,  100, 
and  32,  the  first  remainder,  added,  we  obtain  432  for  tlie  true  remainder 
(Art.  66).     Hence  each  soldier  receives  9|f f  pounds. 

59.  Rule.  —  Cut  off  tlie  ciphers  from  the  divisor,  and  Oie  same 
number  of  figures  from  the  right  of  the  dividend. 

Then  divide  Vie  remaining  fgures  of  the  dividend  by  the  remaining 
fgures  of  the  div'iaor. 

.57.  How  do  you  divide  by  10?  IIow  docs  it  appear  thut  this  divides  tho 
nninhcr  hy  10  7 — TiS.  How  do  you  divide  l)y  COO  in  the  oxampk- ?  How 
docs  it  appear  that  this  divides  tho  number  ?  —  59.  The  gcuciul  rule  1 


QUESTIONS   INVOLVING   FRACTIONS.  57 

J^OXE.  —  When  by  the  operation  there  is  a  last  remainder,  to  it  must  be 
annexed  the  figures  cut  ott'  from  the  dividend  to  foini  the  true  remainder. 
Sliould  there  be  no  h\st  remainder,  then  tiie  signiticant  figures,  if  any,  cut 
ott"  from  the  dividend,  will  form  the  true  remainder. 

Examples  for  Practice. 

Quotients.       Rem, 

2.  Divide  3594  by  80.             44  74 

3.  Divide  79872  by  240.           332  192 

4.  Divide  467153  by  700.          667  253 

5.  Divide  13112297  by  8900.  2597 

6.  Divide  71897654325  by  700000000.  497654325 

7.  Divide  3456789123456787  by  990000.  306787 

8.  Divide  967231731328000  by  1020000000.  411328000 
9.'  Divide  33166405115000  by  1600000000.  5115000 

10.  Divide  18191618562300  by  1000000000.     618562300 

11.  Divide  476G6G6000000  by  55550000000.   44916000000 


QUESTIONS    INVOLVING    FRACTIONS. 

60.  If  a  unit  or  individual  thing  is  divided  into  equal  parts, 
each  of  the  parts  is  called  a  Fraction  of  the  number  or  thing 
di\ided.     Hence 

A  Fraclioil  is  one  or  more  equal  parts  of  a  unit. 

Illustra  tions.  —  1 .  When  any  number  or  thing  is  divided  into 
two  equal  part£,  one  of  the  parts  is  called  one  half,  and  is  •written 
thus :  ^. 

2.  When  any  number  or  thing  is  divided  into  three  equal  parts,  one 
of  tlie  pftrts  is  called  one  third  (^-)  ;  ttco  of  the  parts  are  called  two 
thirds  (f). 

3.  When  any  number  or  thing  is  divided  into  four  equal  parts, 
one  of  the  parts  is  called  one  fourth  (^)  ;  three  of  the  parts,  three 
JourUis  (|). 

4.  When  any  number  or  thing  is  divided  into  five  equal  parts,  one 
of  the  parts  is  called  one  fifth  (l)  ;  tivo  parts,  two  fifths  (|)  ;  three 
parts,  three  fifths  (I)  ;  and  four  parts,  four  fifths  (-1). 

60.  What  is  a  fraction  ?  What  is  meant  by  one  half  of  any  number  or 
thing  ?  How  is  it  written  ?  Wliat  is  meant  by  one  third,  and  how  is  it 
M'ritten  ?  What  by  one  fourth,  and  how  written  ?  What  by  one  fifth,  and 
Iiow  written  1  What  by  four  fifths,  and  how  written  ?  How  do  you  find  one 
half  of  any  number  1  How  one  third  ?  How  one  fourth  1  &c.  How  many 
halves  make  a  whole  one  ?  How  many  thirds  ?  How  many  fourths  'i  How 
many  fifths  1 


68  QUESTIONS   INVOLVING   FRACTIONS. 

5.  When  any  number  or  thing  is  divided  into  six  equal  parts 
what  is  one  of  the  parts  called  ?     Two  parts  ?     Five  parts  ? 

6.  When  a  number  or  thuig  is  divided  into  7  equal  parts, 
what  is  1  part  called  ?  2  parts  ?  3  parts  ?  4  parts  ?  5  parts  ? 
6  parts  ? 

7.  When  a  number  or  thing  is  divided  into  9  equal  parts, 
what  is  1  part  called  ?  2  parts  ?  4  parts  ?  5  parts  ?  7  parts  ? 
8  parts  ? 

8.  What  is  1  Aa//- of  4  ?  Of  8?  Of  IG?  Of  20  ?  Of  28? 
Of  32  ? 

9.  What  is  1  third  of  9  ?  Of  12  ?  Of  15  ?  Of  27  ?  Of  30  ? 
Of  36  ?     Of  60  ? 

10.  What  is  1  fourth  of  8  ?  Of  16?  Of  20?  Of  24?  Of 
40?     Of  48?     Of  100? 

11.  What  is  1  Jifth  of  10?  Of  25  ?  Of  30  ?  Of  35  ?  Of 
45  ?     Of  50  ?     Of  55  ?     Of  65  ? 

12.  What  is  1  sixth  of  12?  Of  18?  Of  30?  Of  42?  Of 
60  ?     Of  72  ?     Of  90  ? 

13.  How  many  fourths  in  1  apple  ? 

14.  How  many  fourths  in  2  apples?  In  3  apples?  In  8 
apples?     In  16  apples? 

15.  How  many  fifths  in  1  barrol  of  flour  ?  In  3  barrels  ?  In 
5  baiTels  ?     In  7  barrels  ?     In  9  barrels  ? 

16.  How  many  sixths  in  1  bu-hel  of  wheat?  In  4  bushels  ? 
In  7  bushels  ?     In  9  bushels  ?     In  12  bushels  ? 

17.  James  owns  3  fifths  of  a  kite,  and  his  brother  Thomas  the 
remainder.     How  many  fifths  does  Thomas  own  ? 

Illustration.  —  Since  there  are  5  fifths  in  the  kite,  if  James 
owns  3  fifths,  there  will  remain  for  Thomas  5  fifths  (|)  less  3  filths 
(f )  =  2  fifths.  Aus.  2  fifths. 

18.  From  a  load  of  hay  I  sold  4  sevenths;  how  many  sev- 
enths remain  ? 

19.  John  Jones  found  a  large  sum  of  money  ;  he  gave  5 
eighths  of  it  to  the  poor  of  the  parish ;  how  much  did  he  reserve 
for  himself? 

20.  John  Smith  gave  2  ninths  of  his  farm  to  liis  son,  3  ninths 
to  his  daughter,  and  the  remainder  to  his  wife  ;  how  many  ninths 
did  his  wife  receive  ? 

Illustration.  —  Since  he  gave  2  ninths  (5)  to  his  son,  and  3 
ninths  (f)  to  his  (kui^^^htcr.  he  g.ive  tlu-m  both  |  -|-  |-  =  | ;  and 
since  there  arc  0  ninths  Q)  in  the  farm,  his  witc  must  have  re- 
ceived ^  —  ^  =  i-  Ans.  ^. 


QUESTIONS   INVOLVING   FRACTIONS.  59 

21.  In  a  certain  school  -^j  of  the  pupils  study  grammar,  -f^ 
study  arithmetic,  y^j-  geograpiiy,  and  the  remainder  philosophy. 
What  part  of  the  school  study  philosophy  ? 

22.  J.  Dow  spends  ^  of  his  time  in  reading,  ^  in  labor,  and  ^ 
in  visiting.  How  large  a  portion  of  his  time  remains  for  eating 
and  sleeping  ? 

23.  If  a  yard  of  cloth  cost  8  dollars,  what  cost  :|-  of  a  yard  ? 
What  cost  f  of  a  yard  ? 

Illustration.  —  If  1  yard  cost  8  dollars,  ^  of  a  yard  will  cost  \  of 
8  dollars  =  2  dollars ;  and  if  ^  of  a  yard  cost  2  dollars,  |  will  cost 
three  times  as  much,  or  6  dollars.  Ans.  6  dollars. 

24.  If  an  acre  of  land  cost  21  dollars,  what  will  |-  of  an  acre 
cost  ?     What  will  |-  cost  ? 

25.  When  96  cents  are  paid  for  a  bushel  of  rye,  what  cost  \^ 
of  a  bushel ? 

26.  If  I  of  a  barrel  of  flour  cost  2  dollars,  what  cost  f  of  a 
barrel ? 

Illustration.  —  If  |  cost  2  dollars,  |  will  cost  4  times  2  dollars 
=  8  dollai-s.  Ans.  8  dollars. 

27.  If  ^  of  an  acre  of  land  cost  24  dollars,  what  will  |-  of  an 
acre  cost  ? 

28.  If  ^  of  a  hogshead  of  molasses  cost  1 1  dollars,  what  will  a 
hogshead  cost  ? 


o' 


29.  If  I  of  an  acre  of  land  cost  21  dollars,  what  cost  |  of  an 
acre  ?     What  cost  an  acre  ?     What  cost  10  acres  ? 

Illustration.  —  If  |  cost  21  dollars,  I  will  cost  \  of  21  dollars, 
and  4  of  21  dollars  is  3  dollars;  and  |  will  cost  8  times  3  dollars,  or 
24  dollars,  and  10  acres  will  cost  10  tunes  24  dollars,  or  240  dollars. 

Ans.  240  dollars, 

30.  If  y^j.  of  a  hogshead  of  sugar  cost  18  dollars,  what  costs  1 
hogshead  ?     What  cost  4  hogsheads  ? 

31.  If  I  of  a  barrel  of  apples  cost  150  cents,  what  costs  a 
barrel  ?     What  cost  10  barrels  ? 

32.  When  49  dollars  are  paid  for  -/j-  of  a  ton  of  potash,  what 
must  be  paid  for  2  tons  ? 

33.  How  many  half-barrels  of  flour  are  there  in  2  and  a  half 
(2^L)  barrels  ? 

Illustration.  —  Since  1  barrel  contains  2  halves,  2  barrels  will 
contain  2  times  2  halves,  or  4  halves,  and  the  1  half  added  makes  5 
halves.  Ans.  |. 

34.  How  many  half-bushels  in  4^  bushels  of  oats  ?  In  5^ 
bushels  ?     In  7^  bushels  ?     In  9^  bushels  ? 


60  QUESTIONS  INVOLVING   FRACTIONS. 

35.  How  many  eighths  of  a  dollar  in  2^  dollars  ?     In  4|  do! 
lars  ?     In  7|  dollars  ?     In  9^  dollars  't     In  12^  dollars  ? 

36.  How  many  tenths  of  an  ounce  in  4y3g  ounces  ?  In  5/^ 
ounces  ?     In  Sy^j-  ounces  ?     In  lOy^jj  ounces  'i 

37.  How  many  barrels  of  wine  in  6  half  (|)  barrels  ? 

Illustration.  —  Since  it  takes  2  halves  to  make  one  whole  one, 
there  will  be  as  many  whole  barrels  in  6  half-barrels  as  2  is  contained 
times  in  6,  or  3  barrels.  Ans.  3  barrels. 

38.  How  many  firkins  of  butter  in  f  firkins  ?     In  ^-  firkins  .^ 

39.  How  many  whole  numbers  in  -U*.  ?     jj,  .i_5.  p     j^  2^  p 

40.  How  many  whole  numbers  in  ^^-  ?    In  f  ?    In  s-^-  ?    In  ^g^-  ? 

41.  If  a  skein  of  silk  is  worth  3^  cents,  what  are  6  skeins 
worth  ? 

Illustration.  —  If  1  skein  is  worth  31  cents,  6  skeins  are  worth 
6  times  as  much ;  6  times  3^  are  equal  to  6  times  3  and  6  times  ^  ; 
6  times  3  =  18;  6  times  l  =  |  =  3  ;  18  cents  -|-  3  cents  =  21  cents. 

Ans.  21  cents. 

42.  Bought  one  pair  of  boots  for  6j  dollars  ;  what  must  I  pay 
for  4  pairs  ?     For  8  pairs  ?     For  10  pairs  ?     For  12  pairs  ? 

43.  Paid  12^  cents  for  one  pound  of  cloves  ;  what  wiU  G 
pounds  cost?     10  pounds?     12  pounds? 

44.  If  one  pound  of  butter  is  worth  12  cents,  what  are  4^ 
pounds  worth  ? 

Illustration.  —  If  1  pound  is  worth  12  cents,  4^  pounds  are 
worth  4^  times  as  much;  4^  times  12  cents  are  equal  to  4  times  12 
cents  and  \  of  12  cents;  4  "times  12  cents  are  48  cents,  and  \  of  12 
cents  is  6  cents  ;  48  cents  and  6  cents  are  54  cents.      Ans.  54  cents. 

45.  "When  lard  is  sold  for  9  cents  per  pound,  wliat  must  be 
paid  for  7^  pounds  ?     For  8^  pounds  ?     For  9^  pounds  ? 

46.  Bought  1  pound  of  coffee  at  16  cents  ;  what  Avill  5^  pounds 
cost  ?     3^  pounds  ?     5|  pounds  ?     6|  pounds  ? 

47.  If  1  yard  of  cloth  is  worth  20  cents,  what  is  the  value  of 
1  ^  yards  ?     12^  yards  ?     8^  yards  ?     11^  yards  ? 

48.  If  1^  bushels  of  corn  cost  120  cents,  what  will  1  bushel 
cost? 

Illustration.  —  1^  bushels  =  ■§  bushels.  Now,  if  I  cost  120 
cents,  \  will  cost  \  of  1 20  eents,  or  40  cents ;  and  |,  or  a  whoki  bushel, 
will  cost  2  times  40  cents,  or  80  cents.  Ans.  80  cents. 

49.  If  2|  pounds  of  coffee  cost  60  cents,  wliat  will  1  pound 
cost? 


CONTRACTIONS   IN   MULTIPLICATION.  ©1 

Illustration.  —  2|  pounds  =  ^  pounds.  If  ig^  cost  60  cents,  \ 
will  cost  ^^  of  (JO  cents,  or  5  cents  ;  and  ^,  or  a  pound,  will  cost  5  times 
5  cents,  or  25  cents.  Ans.  25  cents. 

50.  How  many  times  will  GO  contain  2f  ? 

51.  Paid  54  dollars  for  1^  barrels  of  oil ;  what  cost  one  bar- 
rel? Ans.  7  dollars. 

52.  How  many  times  is  7^  contained  in  54  ? 

53.  How  many  cords  of  wood  at  5^  dollars  per  cord,  can  be 
bought  for  (SO)  dollars  ? 

54.  How  many  times  will  66  contain  5^  ? 

55.  Gave  forty  dollars  for  6f  yards  of  broadcloth  ;  what  cost 
1  yard  ? 

56.  How  many  times  is  6§  contained  in  40  ? 

57.  The  distance  between  two  places  is  110  rods.  I  wish  to 
divide  this  distance  into  spaces  of  5^  rods  each.  Required  the 
number  of  spaces  ? 


CONTRACTIONS.* 
CONTRACTIONS  IN  MULTIPLICATION. 

61.    To  multiply  by  25. 

Ex.  1.   MuUiply  876581  by  25.  Ans.  21914525. 

orERATiON.  ^^Q  multiply  by  100,  by  annex- 

4)87658100  ing  two  ciphers   to   the   multipli- 

n  t  t\  1   J  c  n  c   -r>     1     i         cand :    and   since    25,   the   multi- 
2  19  14  5  2  5  Product.       pu^^.;;^  ^^^1^  ^,^^  ^^^'^^^/^  ^^  ^^0, 

we  divide  by  4  to  obtain  the  true  product. 
Rule.  —  Annex  two  ciphers  to  the  muUipUcand,  and  divide  it  hy  4. 

*  If  the  principles  on  wliich  these  contractions  depend  are  considered 
too  difficuh  for  tlio  3'oiiti2;  pupil  to  understand  at  diis  stage  of  his  pronrress, 
tliey  may  be  omitted  for  the  present,  and  attended  to  when  he  is  further 
advanced. 

61.  The  rule  for  multiplying  by  25  ?     The  reason  for  the  rule  1 
6 


62  CONTRACTIONS  IN  MULTIPLICATION. 

Examples  for  Pkactice. 

2.  Multiply  7G589GJ8  by  25.  Ans.  1914741450. 

3.  Multiply  507898717  by  25.  Ans.  141974G7925. 

4.  Multiply  12345G789  by  25.  Ans.  3086419725. 

62.    To  multiply  by  .33^. 

Ex  1.   Multiply  87G789G3  by  33J'.  Ans.  2922G32100. 

orERATioN.  We   multiply  by  100,  as   be- 

3  )  8  7  G  7  8  9  6  3  0  0  fore  ;  and  since  33^- ,  tlie  multi- 

plier,  is  only  one  third  of  100, 

2922632100  Product.        wo  divide  by  3  to  obtain  the  true 

product. 

KuLE.  —  Annex  two  ciphers  to  the  muldplicand,  and  divide  it  by  3. 

Examples  for  Practice. 

2.  Multiply  35G789541  by  33^.  Ans.  11892984700. 

3.  Multiply  871132182  by  33^.  Ans.  29037739400. 

4.  Multiply  583G47912  by  33^.  Aus.  19454930400. 

03,   To  multiply  by  125. 

Ex.  1.   Multiply  789G538  by  125.  Ans.  9870G7250. 

OPERATION.  "We  multiply  by  1000,  by  an- 

8)7896538000  nexing   three    ciphers    to    the 

multiplicand;  and  since  125, the 
987067250  Product,      multiplier,  is  only  one  eighth  of 
1000,  we  divide  by  8  to  obtain  the  true  product. 

KuLE.  —  Annex  three  ciphers  to  the  multiplicand,  and  divide  it  by  8. 

Examples  for  Practice. 

2.  Multiply  7965325  by  125.  Ans.  995665625. 

3.  Multiply  1234567  by  125.  Ans.  15   '  20875. 

4.  Multiply  3049862  by  125.  Ans.  381232750. 

f)2.  TIic  nilo  for  mnlti|)lvin<,'  hv  .^.3J  ?    The  rc.ison  for  this  rule?  —  63.  Tho 
rule  for  multiply iujj  by  125  1     The  reiisou  for  the  rule  1 


CONTRACTIONS   IN   DIVISION.  63 

61.    To  multiply  by  any  number  of  9's. 

Ex.  1.  Multiply  4789G53  by  9D999.     Ans.  478960510347. 

oPEnATiON.  By  adding  1  to  any  number 

478965300000  conijiosed  of  nines,  we  obtain  a 

4  7  8  9  6  5  3  number  expressed  by  1  -vviili  as 

many  cipher^s  annexed  as  there 


478960510347  Product,     are    nines    in   the    number   to 

which  the  1  is  added.  Thus, 
999  -}-  1  =  1000.  Therefore,  annexing  to  the  multiplicand  as  many 
ciphers  as  tliere  are  nines  in  the  multiplier  is  the  same  as  multiplviiifr 
the  immber  by  a  nmltiplier  too  large  by  1,  and  subtracting  the  number 
to  be  multiplied  from  this  enlarged  product  will  give  the  true  product. 

Rule.  —  Annex  as  many  ciphers  to  the  muldpHcand  as  there  are  9's 
in  the  multiplier,  and  from  this  number  subtract  the  number  to  be  mul- 
tiplied. 

Examples  for  Practice. 

2.  Multiply  1234567  by  999.  Ans.  1233332433. 

3.  Multiply  876543  by  999999.  Ans.  876542123457. 

4.  Multiply  999999  by  999999.  Ans.  999998000001. 


CONTRACTIONS  IN  DIVISION. 

65.   To  divide  by  25. 

Ex.  1.  Divide  1234567  by  25.  Ans.  49382  iVij- 

OPERATION.  Multiplying  the  dividend  by  4  makes 

1234567  it  four  times  as  great ;  therefore,  to  ob- 

4  tain  the  true  quotient,  we  must  divide  by 

100,  a  divisor  four  times  as  great  as  the 

4  9  3  8  2| 6  8   Quotient.      true  one.     This  we  do  by  cutting  ofl'  two 

figures  on  the  right. 

EuLE.  —  Multiply  the  dividend  by  4,  and  divide  the  product  by  100. 

Examples  for  Practice. 

2.  Divide  9876525  by  25.  Ans.  395061. 

3.  Divide  1378925  by  25.  Ans.  55157. 

4.  Divide  899999  by  25.  Ans.  35999/^V 

64.  The  rule  for  mnltiplvinjj  bv  any  number  of  9's  ?     The  reason  for  iho 
rule  1  —  65.  The  rule  for  dividing  by  25  ?     The  reason  for  the  rule  1 


64 


CONTRACTIONS  IN  DIVISION. 


66.    To  divide  by  33^. 

Ex.  1.  Divide  6789543  by  33'-. 


Ans.  20368G/^V 


OPERATION. 

6789543 
3 

203686|29  Quotient. 


Multiplying  the  dividend  by  3  makes 
it  three  times  as  great ;  therefore,  to 
obtain  the  true  quotient,  we  must  divide 
by  100,  a  divisor  three  times  as  great 
as  the  true  one.  This  is  done  by  cut- 
ting off  two  figures  on  the  right. 


Rule.  —  Multiply  iJie  dividend  hy  3,  and  divide  the  product  hy  100. 


Examples  for  Practice. 


2.  Divide  987654321  by  33^. 

3.  Divide  8712378  by  33^. 

4.  Divide  4789536  by  33^. 

5.  Divide  89676  by  33^. 

6.  Divide  17854  by  33^. 

67.   To  divide  by  125. 

Ex.  1.  Divide  9874725  by  125. 


Ans.  29629629-j-6(jV 

Ans.  261371^3jV 

Ans.  143686i§^. 

Ans.  2690^258^. 

Ans.  535j-^^2j5-. 


Ans.  78997-jV 


OPERATION. 

98747  25 
8 

78997|800  Quotient. 


Multiplying  the  dividend  by  8  makes 
it  eight  times  as  great ;  therefore,  to  ob- 
tain the  true  quotient,  we  must  divide 
by  1000,  a  divisor  eight  times  as  great 
as  the  true  one.  We  do  this  by  cut- 
ting oflf  three  figures  on  the  right. 


Rule.  —  Multiply  the  dividend  by  8,  arid  divide  the  product  by  1000. 


Examples  for  Practice. 


2.  Divide  1728125  by  125. 

3.  Divide  478763250  by  125. 

4.  Divide  591234875  by  125. 

5.  Divide  489648  by  125. 

6.  Divide  836184  by  125. 


Ans.  13825. 

Ans.  3830106. 

Ans.  4729879. 
Ans.  3917-j-'aVjy. 
Ans.  6689-i*gVV 


66.  The  rule  for  dividing:  by  33$  ?     The  reason  for  the  rule? — 67.  The 
rule  for  dividing  by  125  ?     The  reason  for  the  rule  f 


inSCELLANEOUS   EXAJIPLES.  65 

MISCELLANEOUS    EXAMPLES, 

INVOLVING    TUE    FOREGOING    RULE3. 

1.  A  bought  73  hog-heads  of  molasses  at  29  dollars  per 
hogshead,  and  sold  it  at  37  dollars  per  hogshead  ;  what  did  lie 
gain  ?  Alls.  584  dollars. 

2.  B  bought  896  acres  of  wild  land  at  15  dollars  per  acre, 
aud  sold  it  at  43  dollars  per  acre  ;  what  did  he  gain  ? 

Ans.  25088  dollars. 

3.  N.  Gage  sold  47  bushels  of  corn  at  57  cents  per  bushel, 
which  cost  him  only  37  cents  per  bushel ;  how  many  cents  did 
he  gain  ?  Ans.  940  cents. 

4.  A  butcher  bought  a  lot  of  beef  weighing  765  pounds  at  11 
cents  per  pound,  and  sold  it  at  9  cents  per  pound  ;  how  many 
cents  did  he  lose  .''  Ans.  1530  cents. 

5.  A  taverner  bought  29  loads  of  hay  at  17  dollars  per  load 
and  76  cords  of  wood  at  5  dollai's  a  cord ;  what  was  the  amount 
of  the  hay  and  the  wood  ?  Ans.  873  dollars. 

6.  Bought  17  yards  of  cotton  at  15  cents  per  yard,  46  gallons 
of  molasses  at  28  cents  per  gallon,  16  pounds  of  tea  at  76  cents  a 
pound,  and  107  pounds  of  coffee  at  14  cents  a  pound  ;  what  was 
the  amount  of  my  bill  ?  Ans.  4257  cents. 

7.  A  man  traveled  78  days,  and  each  day  he  walked  27  miles; 
what  was  the  length  of  his  journey  ?  Ans.  2106  miles. 

8.  A  man  set^  out  from  Boston  to  travel  to  New  York,  the 
distance  being  223  miles,  and  walks  27  miles  a  day  for  6  days 
in  succession  ;  what  distance  remains  to  be  traveled  ? 

Ans.  61  miles. 

9.  "What  cost  a  farm  of  365  acres  at  97  dollars  per  acre  ? 

Ans.  35405  dollars. 

10.  Bought  376  oxen  at  36  dollars  per  ox,  169  cows  at  27 
dollars  each,  765  sheep  at  4  dollars  per  head,  and  79  elegant 
hories  at  275  dollars  each ;   what  was  paid  for  all  ? 

Ans.  42884  dollars. 

11.  J.  Barker  has  a  fine  orchard,  consisting  of  365  trees,  and 
each  tree  produces  7  barrels  of  apples,  and  these  apples  will 
bring  him  in  market  3  dollars  per  barrel ;  what  is  the  income 
of  the  orchard  ?  Ans   7665  dollars. 

1 2.  J.  Peabody  bought  of  E.  Ames  7  yards  of  his  best  broad- 
cloth at  9  dollars  per  yard,  and  in  payment  he  gave  Ames  a  one 

6* 


66  MISCELLANEOUS  EXAMPLES. 

hundred-dollar  bill  ;   how  many  dollars   must  Ames   return  to 
Peabody?  Ans.  37  dollars. 

13.  Bought  of  P.  Parker  a  cooking-stove  for  31  dollars,  7 
quintals  of  his  best  fish  at  6  dollars  per  quintal,  14  bushels  of  rye 
at  1  dollar  per  bushel,  and  5  mill-saws  at  16  dollars  each ;  in  part 
payment  for  the  above  articles,  I  sold  him  eight  thousand  feet  of 
boards  at  15  dollars  per  thousand ;  how  much  must  I  pay  him  to 
balance  the  account  ?  Ans.  47  dollars. 

14.  In  1  day  there  are  24  hours  ;  how  many  in  57  days  ? 

Ans.  13G8  hours. 

15.  In  one  pound  avoirdupois  weight  there  are  16  ounces ;  how 
many  ounces  are  there  in  369  pounds  ?  Ans.  5904  ounces. 

16.  In  a  square  mile  there  are  640  acres  ;  how  many  acres 
are  there  in  a  town  which  contains  89  square  miles  ? 

Ans.  56960  acres. 

17.  What  cost  78  barrels  of  apples  at  3  dollars  per  barrel  ? 

Ans.  234  dollars. 

18.  Bought  500  barrels  of  flour  at  5  dollars  per  barrel,  47  hun- 
dred weight  of  cheese  at  9  dollars  per  hundred  weight,  and  15 
barrels  of  salmon  at  17  dollars  per  barrel ;  what  was  the  amount 
of  my  purchase  ?  Ans.  3178  dollars. 

19.  Bought  760  acres  of  land  at  47  dollars  per  acre,  and  sold 
J.  Emery  171  acres  at  66  dollars  per  acre,  J.  Smith  275  acres  at 
37  dollars  per  acre,  and  the  remainder  I  sold  to  J.  Kimball  at  75 
dollars  per  acre  ;  how  much  did  I  gain  by  my  sales  ? 

An.^.  7581  dollars. 

20.  Bought  a  hogshead  of  oil  containing  184  gallons,  at  75  cents 
per  gallon  ;  but  28  gallons  having  leaked  out,  I  sold  tlie  remaiu- 
der  at  98  cents  per  gallon  ;  did  I  gain  or  lose  Ijy  nn-  bargain  ? 

Ans.  1488  cents,  gain. 

21.  Bought  a  quantity  of  flour,  for  which  I  gave  1728  dollars, 
there  being  288  barrels  ;  I  sold  the  same  at  8  dollars  per  barrel ; 
how  much  did  I  gain  ?  Ans.  576  dollars. 

22.  Purchased  a  cargo  of  molasses  for  9212  dollars,  there 
being  196  hogsheads  ;  I  sold  the  same  at  67  dollars  per  hogs- 
head ;  how  much  did  I  gain  on  each  hogshead  ? 

Ans.  20  dollars. 

23.  A  fnrmor  bought  5  yoke  of  oxen  at  87  dollars  a  yoke  ;  37 
cows  at  37  dollars  each;  89  sheep  at  3  dollars  apiece,  lie  sold 
the  oxen  at  98  dollars  a  yoke  ;  for  the  cows  he  received  40  dollars 
each  ;  and  for  the  sheep*  he  had  4  dollars  apiece.  How  nuich  did 
he  -aiii  by  his  trade?  Ans.  255  dollars. 


MISCELLANEOUS  EXAMPLES.  67 

24.  The  sum  of  two  numbers  is  5482,  and  the  smaller  number 
is  1962  ;  what  is  the  ditierence  ?  Ans.  3520. 

25.  The  diiference  between  two  numbers  is  125,  and  the 
smaller  number  is  1482  ;  what  is  the  greater?  Ans.  1607. 

26.  The  difference  between  two  numbers  is  1282,  and  the 
greater  number  is  6958  ;  what  is  the  smaller?  Ans.  5676. 

27.  If  the  dividend  is  21775,  and  the  divisor  871,  what  is 
the  quotient  ?  Ans.  25. 

28.  If  the  quotient  is  482,  and  the  divisor  281,  what  is  the 
dividend  ?  Ans.  135442. 

29.  If  144  inches  make  1  square  foot,  how  many  square  feet 
in  20736  inches  ?  Ans.  144  feet. 

30.  An  acre  contains  160  square  rods  ;  how  many  square  rods 
in  a  farm  containing  769  acres .''  Ans.  123040  I'ods. 

31.  A  gentleman  bought  a  house  for  three  thousand  forty- 
seven  dollars,  and  a  carriage  and  span  of  horses  for  five  hundred 
seven  dollars.  He  paid  at  one  time  two  thousand  seventeen 
dollars,  and  at  another  time  nine  hundred  seven  dollars.  How 
much  remains  due  ?  Ans.  630  dollars. 

32.  The  erection  of  a  factory  cost  68,255  dollars  ;  supposing 
this  sum  -to  be  divided  into  365  shares,  what  is  the  value  of 
each  ?    •  Ans.  187  dollars. 

33.  Bought  two  lots  of  wild  land  ;  the  first  contained  144 
acres,  for  which  I  paid  12.  dollars  per  acre  ;  the  second  contained 
108  acres,  which  cost  15  dollars  per  acre.  I  sold  both  lots  at 
18  dollars  per  acre  ;  what  was  the  amount  of  gain  ? 

Ans.  1188  dollars. 

34.  Sold  17  cords  of  oak  wood  at  6  dollars  per  cord,  36  cords 
of  maple  at  3  dollars  per  cord,  and  29  cords  of  walnut  at  7  dol- 
lars per  cord.     AVliat  was  the  amount  received  ? 

Ans.  413  dollars. 

35.  Daniel  Bailey  has  a  fine  farm  of  300  acres,  which  cost 
him  73  dollars  per  acre.  He  sold  83  acres  of  this  farm  to  Minot 
Thayer,  for  97  dollars  per  acre  ;  42  acres  to  J.  Russel,  for  87 
dollars  per  acre  ;  75  acres  to  J.  Dana,  at  75  dollars  per  acre  ; 
and  the  remainder  to  J.  "Webster,  at  100  dollars  per  acre.  What 
was  his  net  gain  ?  Ans.  5430  dollars. 

36.  J.  Gale  purchased  17  sheep  for  3  dollars  each,  19  cows 
at  27  dollars  each,  and  47  oxen  at  57  dollars  each.  He  sold  his 
purchase  for  3700  dollars.     What  did  he  gain  ? 

Ans.  457  dollars. 

37.  Purchased  17  tons  of  copperas  at  32  dollars  per  ton.  I 
sold  7  tons  at  29  dollars  per  ton,  8  tons  at  36  dollars  per  ton, 


68  MISCELLANEOUS  EXAMPLES. 

and  the  remainder  at  25  dollars  per  ton.     Did  I  gain  or  lose,  and 
how  mucli  ?  Ans.  3  dollars,  loss. 

38.  John  Smith  bought  28  yards  of  broadcloth  at  5  dollars 
per  yard  ;  and,  having  lost  10  yards,  he  sold  the  remainder  at  9 
doUiirs  per  yard.     Did  he  gain  or  lose,  and  how  much  ? 

Ans.  22  dollars,  gain. 

39.  Which  is  of  the  greater  value,  386  acres  of  land  at  76 
dollars  per  acre,  or  968  hogsheads  of  molasses  at  25  dollars  per 
hogshead  ?  Ans.  The  land,  by  5136  dollars. 

40.  Bought  of  J.  Low  37  tons  of  hay  at  18  dollars  per  ton. 
I  paid  him  75  dollars,  and  12  yards  of  broadcloth  at  4  dollars 
per  yard.     How  much  remains  due  to  Low  ? 

Ans.  543  dollars. 

41.  A  purchased  of  B  40  cords  of  wood  at  5  dollars  per  cord, 
9  tons  of  hay  at  17  dollars  per  ton,  19  grindstones  at  2  dollars 
apiece,  37  yards  of  broadcloth  at  4  dollars  per  yard,  and  16 
barrels  of  flour  at  6  dollars  per  barrel ;  what  is  tlie  amount  of 
A's  bill  ?  Ans.  635  dollars. 

42.  John  Smith,  Jr.,  bought  of  R.  S.  Davis  18  dozen  of 
National  Arithmetics  at  6  dollars  per  dozen,  23  dozen  of  Mental 
Arithmetics  at  1  dollar  per  dozen,  17  dozen  Family  Bibles  at  3 
dollars  per  copy;  what  was  the  amount  of  the  bill? 

Ans.  743  dollars. 

43.  R.  Hasseltine  sold  to  John  James  169  tons  of  timber  at 
7  dollars  per  ton,  116  cords  of  oak  wood  at  6  dollars  per  cord, 
and  37  cords  of  maple  wood  at  5  dollars  per  cord ;  James  has 
paid  Hasseltine  144  dollars  in  cash,  and  23  yards  of  cloth  at  4 
dollars  per  yard ;  what  remains  due  to  Hasseltine  ? 

Ans.  1828  dollars. 

44.  J.  Frost  owes  me  on  account  375  dollars,  and  he  has  paid 
me  6  cords  of  wood  at  5  dollars  per  cord,  15  tons  of  hay  at  12 
dollars  per  ton,  and  32  bushels  of  rye  at  1  dollar  jier  bu.-heL 
How  much  remains  due  to  me  ?  Ans.  133  dollars. 

45.  Gave  169  dollars  for  a  chaise,  87  dollars  for  a  harness, 
and  176  dollars  for  a  liorsc.  I  sold  the  chaise  for  187  dollars^ 
the  harness  for  107  dollars,  and  the  horse  for  165  dollars. 
"Wiiat  sum  have  I  gained  ?  Ans.  27  dollars. 

46.  Bought  a  farm  of  J.  C.  Bradbury  for  1728  dollars,  for 
which  I  paid  him  75  l)arrels  of  flour  at  6  dollars  prr  barrel,  9 
cords  of  wood  at  5  dollars  a  cord,  17  tons  of  hay  at  2')  dollars 
a  ton,  40  bushels  of  wheat  at  2  dollars  a  bushel,  and  65  bushels 
of  beans  at  3  dollars  a  bushel ;  how  many  dollars  remain  due  to 
Bradbury  ?  Ans.  533  dollars. 


UNITED   STATES   MONEY.  69 


UNITED    STATES    MONEY. 

68.    United  States  Money,  established  by  Congress  in  179G,  is  the 
legal  currency  ot"  the  United  States. 

TABLE. 


10  Mills 

make 

1  Cent, 

marked 

C. 

10  Cents 

(( 

1  Dime, 

(( 

d. 

10  Dimes 

(( 

1  Dollar, 

°(( 

$. 

10  Dollars 

<( 

1  Eagle, 
Dimes. 

Cents. 

1       = 

E. 

Mills. 
10 

Dollars.  1       =  10       =  100 

Eagle.  1         =  10      =         100       =  1000 

1  =  10         =         100      =      1000       =       10000 

Simple  Numbers,  that  is,  numbers  whose  units  are  all  of  a  single 
kind  or  denomination,  have  thus  far  been  made  use  of  in  this 
work. 

The  units  or  denominations  of  United  States  money  increase 
from  right  to  left,  and  decrease  from  left  to  right,  in  the  same 
manner  as  do  the  units  of  the  several  orders  in  simple  num- 
bers ;  and  may,  therefore,  be  added,  subtracted,  multiplied,  and 
divided  in  hke  manner  as  simple  numbers. 

Dollars  are  separated  from  cents  by  a  point  ( . )  called  a  separa- 
trix  or  decimal  point ;  the  first  two  places  at  the  right  of  the  point 
being  cents;  and  the  third  place,  mills.  Thus,  $16,253  is  read, 
sixteen  dollars,  twenty-five  cents,  three  mills. 

Since  cents  occupy  two  places,  the  place  of  dimes  and  of  cents, 
when  the  number  of  cents  is  less  than  10,  a  cipher  must  be 
written  in  the  place  of  dimes  ;  thus,  .03,  .07,  &c. 

The  Coins  of  the  United  States  consist  of  the  double-eagle,  eagle, 
half-eagle,  quarter-eagle,  three  dollai's,  and  dollar,  made  of  gold ; 
the  dollar,  half-dollar,  quarter-dollar,  dime,  half-dime,  and  three- 
cent  piece,  made  of  silver  ;  the  cent  and  two  cent  pieces  are  made 
of  copper. 

Note  1. — Mill  is  from  the  Latin  word  mille  (one  thousand) ;  Cent,  from 
centum  (one  hundred) ;  Dime,  from  a  French  word  signifying  a  tithe  or  tenth. 

68.  "What  is  United  States  money  *  Repeat  the  table  of  United  States 
money.  What  is  a  simple  number  ?  The  denominations  of  United  States 
money  1  How  do  they  increase  from  ri<rht  to  left  ?  How  are  they  added, 
subtracted,  multiplied,  and  divided  ?  How  are  dollars  separated  from  rents 
and  mills  T  Why  must  a  cipher  be  placed  before  cents,  when  the  number  is 
less  than  10?  Why  are  two  places  allowed  for  cents,  while  only  one  is 
allowed  for  mills  ?     Name  the  coins  of  tlie  United  States. 


70  UNITED   STATES   MONEY. 

The  term  Dollar  is  said  to  be  derived  from  the  Danish  word  Daler,  and 
this  from  iJale,  the  name  of  a  town,  where  it  was  first  coined. 

The  symbol  $  represents,  probably,  the  letter  U  written  upon  an  S,  denot- 
ing U.  S-  (United  States.) 

Note  2. — All  the  gold  and  silver  coins  of  the  United  States  are  now  made 
of  one  purity,  nine  parts  of  pure  metal,  and  one  part  alloy.  The  alloy  for 
the  silver  is  pure  copper ;  and  that  for  the  gold,  one  part  copper  and  one  part 
silver.  The  cent  is  now  made  of  95  parts  of  copper  and  5  parts  tin  and 
zinc. 

The  standard  weight,  as  fixed  by  present  laws,  of  the  eagle,  is  2.58  grains, 
Troy;  the  silver  dollar,  41 2i- grains ;  half-dollar,  192  grains;  quarter-dollar, 
96  grains  ;  dime,  38f  grains  ;  half-dime,  19^  grains;  three-cent  piece,  HiVo 
grains  ;  and  the  cent,  new  coinage,  48  grains. 

Note  3.  —  The  currency  of  Canada,  established  in  18.57;  like  that  of  the 
United  States,  has  for  its  unit  the  dollar,  consisting  of  100  cents,  of  10  mills 
each. 

EEDUCTION. 

69.  Reduction  of  United  States  Money  is  changing  the  units  of 
one  of  its  denominations  to  units  of  another,  of  equal  value. 

70.  To  reduce  from  a  higher  to  a  lower  denomination. 

Ex.  1.     Reduce  25  dollars  to  cents  and  mills. 

Ans.  2500  cents,  25000  mills. 

OPERATION. 

2  5  dollars. 
100  We  multiply  the  25  hy  100,  he- 

„  r  n  ,-v        i             cause  100  cents  make  1  dollar:  and 
2  0  0  0  cents.         u:.,i„  *t,„  „;;rvA  u„  ia   v L  ia 


10 


multiply  the  2500  by  10,  because  10 
mills  make  1  cent. 


2  5  0  0  0  mills. 
Or  thus,    2  5  0  0  0  mills. 

Rule.  —  To  reduce  dollars  to  cents,  annex  two  ciphers ;  to  reduce 
dollars  to  mills,  annex  three  ciphers ;  and  to  reduce  cents  to  mills, 
annex  ONE  cipher. 

Note.  —  Dollars,  cents,  and  mills,  expressed  by  a  single  number,  are 
reduced  to  mills  by  merely  removing  the  separating  point;  and  dollars  and 
cents,  by  annexing  one  cipher  and  removing  the  separatrix. 

71.   To  reduce  from  a  lower  to  a  higher  denomination. 

Ex.  1.  Reduce  25000  mills  to  cents  and  dollar?. 

Ans.  2500  cents,  $  25. 

60.  Wiiat  i.s  reduction  of  United  States  Money  ?  —  70.  What  is  the  rule 
for  reducing  dollars  to  cents  and  mills  ?  The  reason  for  the  riile  ?  How  do 
vou  rcdiu'C  dollars  and  cents  to  cents,  or  dollars,  cents,  and  mills,  to  mills  ? 
'riio  reason  for  this  rule  1 


ADDITION.  71 


OPERATION. 


1  0  )25000  mills.  ^y^  ^j^j^^  ^j^^  25000  by  1 0,  because 

100)2500  cents.  1^  mills  makes  1  cent ;  and  divide  the 

*  2500  by  100,  because  100  cents  make 

2  5  dollars.        1  dollar. 

Orthus,  2  510  010  mills. 

Rule. —  To  reduce  inilb  to  cents,  cut  off  one  fgure  on  the  right; 
to  reduce  cents  to  dollars,  2)oint  off  TVfO  figures ;  and  to  reduce  mills  to 
dollars,  point  off  turee  Jigwes. 

Examples  for  Practice. 

1.  Reduce  $  125  to  cents.  Ans.  12500  cents. 

2.  Reduce  $  345  to  mills.  Ans.  345000  mills. 

3.  Reduce  297  mills  to  cents.  Ans.  $  0.297. 

4.  Reduce  2682  mills  to  dollars.  Ans.  $  2.682. 

5.  Reduce  4123  cents  to  dollars.  Ans.  $41.23. 

6.  Reduce  $  156.29  to  cents.  Ans.  15629  cents. 

7.  Reduce  $  16.428  to  mills.  Ans.  16428  mills. 

8.  Reduce  $  9.87  to  mills.  Ans.  9870  mills. 

ADDITION. 

72.  Rule. — Write  dollars,  cents,  and  mills,  so  that  units  of  the 
same  denomination  shall  stand  in  the  same  column. 

Add  as  in  addition  of  simple  numbers,  and  place  the  separating  point 
directly  under  that  above. 

Proof.  —  The  proof  is  the  same  as  in  addition  of  simple  num- 
bers. 

Examples  for  Practice. 


1. 

2. 

3. 

4. 

S    cts.  m. 

$    cts.  m* 

$    cts.  m. 

$        cts. 

4  5.2  4  3 

7  5.6  4  3 

1  6.7  0  5 

1  4  7.8  6 

1  3.8  9  6 

1-6.8  9  7 

1  4.0  0  3 

7  8  9.5  8 

9  3.5  1  6 

4  3.8  1  6 

1  8.7  1  9 

4  9  6.3  7 

5  2.3  4  3 

5  8.3  1  3 

9  7.0  0  9 

9  1  1.3  4 

Ans. 

2  0  4.9  9  8 

1  9  4.6  6  9 

1  4  6.4  3  6 

2  3  45.15 

71.  What  is  the  rale  for  reducing  mills  to  cents  ?  For  cents  to  dollars? 
Mills  to  dollars?  The  reason  for  each? — 72.  How  must  the  numbers  be 
written  down  in  addition  of  United  States  money  1  How  added  1  How 
pointed  off  ?     The  rule  ? 


72  UNITED   STATES   MONEY. 


5. 

6. 

7. 

8. 

S     ct8.  m. 

S    cts.  m. 

$    cts.  m. 

$    cts   m. 

7  8  G.7  1  3 

8  7.0  5  9 

9  1.7  6  3 

7  8  6.7  1  3 

1  7  G.O  7  1 

3  7.8  1  0 

8  4.1  6  1 

3  4  5.6  7  8 

5  G  7.8  1  9 

8  1.4  7  5 

1  0.0  7  0 

9  0  7.0  1  7 

1  2  3.4  5  6 

4  0.0  7  8 

5  3.6  1  5 

8  6  1.0  9  0 

7  8  y.o  1  2 

2  1.15  6 

8  1.1  7  6 

1  2  3.4  7  6 

3  4  5.6  7  8 

8  1.1  7  7 

3  2.8  1  7 

9  8  7.0  1  6 

9  0  1.2  3  4 

3  3.6  2  1 

5  3.1  9  6 

3  4  5.7  0  5 

7  1  8.9  0  5 

2  8.0  9  3 

4  1.5  7  0 

3  5  7.0  9  1 

9.  Bought  a  coat  for  $17.81,  a  vest  for  $3.75,  a  pair  of 
pantaloons  for  $  2.87,  and  a  pair  of  boots  for  $  7.18  ;  what  was 
the  amount?  Ans,  ij  31.61. 

10.  Sold  a  load  of  wood  for  seven  dollars  six  cents,  five 
bushels  of  corn  for  four  dollars  seventy-five  cents,  and  seven 
bushels  of  potatoes  for  two  dollars  six  cents ;  what  was  received 
for  the  whole?  "    Ans.  $  13.87. 

11.  Bought  a  barrel  of  flour  for  $6.50,  a  box  of  sugar  for 
$  9.87,  a  ton  of  coal  for  $  12.77,  and  a  box  of  raisins  for  $  2.50  ; 
what  was  paid  for  the  vai'ious  articles?  Ans.  $31.64. 

12.  Paid  $4.62  for  a  hat,  $9.75  for  a  coat,  $5.75  for  a  pair 
of  boots,  and  $1.50  for  an  umbrella;  what  was  paid  lor  the 
whole?  Ans.  $21.62. 

13.  A  grocer  sold  a  pound  of  tea  for  $0,625;  4  pounds  of 
butter  for  $  0.75 ;  4  dozen  of  lemons  for  $  0.875  ;  9  pounds  of 
sugar  for  S  0.80  ;  and  3  pounds  of  dates  ibr  $  0.375.  What  was 
the  amount  of  the  bill  ?  '  Ans.  $  3.42c. 

14.  A  student  purchased  a  Latin  grammar  for  $0.75,  a  Virgil 
for  $3.75,  a  Greek  lexicon  for  $  4.75,  a  Homer  for  $  1.25.  an 
English  dictionary  for  $  3.75,  and  a  Greek  Testament  for  $0.75  ; 
what  was  the  amount  of  the  bill?  Ans.  $  15. 

15.  Bought  of  J.  TI.  Carleton  a  China  tea-=et  for  ten  dollars 
eighty-two  cents,  a  dining-set  for  nine  dollars  sixty-two  cents 
five  mills,  a  solar  lamp  for  ten  dollars  fifty  cents,  a  pair  of  va>ea 
for  four  dollars  sixty-two  cents  five  mills,  and  a  set  of  silver 
epoons  for  twelve  dollars  seventy-five  cents  ;  what  did  the  whole 
cost  ?  Ans.  $  48  32. 

16.  Bought  three  hundred  weight  of  beef  at  seven  dollars 
Fcven  cents  ])cr  hundred  weight,  four  cords  of  wood  at  six  dollars 
four  cents  per  cord,  and  a  clice-c  for  three  dollars  nine  cents  ; 
what  was  the  amount  of  the  bill?  Ans.  $  48.46. 


SUBTRACTION.  73 

SUBTRACTION. 

73 1     Rule.  —  Write   the  several  denominations  of  the  subtrahend 
under  the  corresponding  ones  of  the  minuend. 

Subtract  as  in  subtraction  of  simple  numbers,  and  place  the  separatrix 
directly  under  that  above. 

Proof.  —  The  proof  is  the  same  as  in  subtraction  of  simple 
numbers. 

Examples  for  Pkactice. 

1.  2.  3.  4. 

$    cts.  m.  $       cts.  $     cts.  m.  $       cts. 

Min.        6  1.5  8  5  4  7  1.8  1  1  5  6.0  0  3  1  4  1.7  0 

Sub.  19.1  9  7  1  5  8.1  9  1  9.0  0  9  9  0.9  1 


Rem.        4  2.3  8  8  3  1  3.G  2  1  3  6.9  9  4  5  0.7  9 


5.  6.  7.  8. 

$    cts.  m.  $    cts.  m.  $      cts  m.  $          cts.  m. 

From      7  1.8  6  1  9  1.0  7  1  8  1  5.7  0  1  1  0  7  8  1.3  0  3 

Take       1  9.1  9  7  1  9.0  9  5  9  0.8  0  3  9  9  9  9.0  9  7 


9.  From  $  71.07  take  $  5.09.  Ans.  $  65.98. 

10.  From  $  100  take  $  17.17.  Ans.  $  82.83. 

11.  From  one  hundred  dollars  there  were  paid  to  one  man 
seventeen  dollars  nine  cents,  to  another  twenty-three  dollars 
eight  cents,  and  to  another  thirty-three  dollars  twenty-five  cents ; 
how  much  cash  remained  ?  Ans.  $  26.58. 

12.  From  ten  dollars  take  nine  mills.  Ans.  $  9.991. 

13.  A  lady  went  "  a  shopping,"  her  mother  having  given  her 
fifty  dollars.  She  purchased  a  dx*ess  for  fifteen  dollars  seven 
cents  ;  a  shawl  for  eleven  dollars  ten  cents  ;  a  bonnet  for  seven 
dollars  nine  cents ;  and  a  pair  of  shoes  for  two  dollars.  How 
much  money  had  she  remaining?  Ans.  $  14.74. 

14.  From  one  hundred  dollars  there  were  taken  at  one  time 
thirty-one  dollars  fifteen  cents  seven  mills ;  at  another  time, 
seven  dollars  nine  cents  five  mills  ;  at  another  time,  five  dollars 
five  cents ;  and  at  another  time,  twenty-two  dollars  two  cents 
seven  mills.     How  much  cash  remained  of  the  hundred  dollars  ? 

Ans.  $34,671. 

73.  How  do  you  write  down  the  numbers  in  subtraction  of  United  States 
money  1     How  subtract  ?     How  pointed  off?     The  rule  ? 
7 


74  UNITED   STATES  MONEY. 

MULTIPLICATION. 

74.     Rule.  —  Multiply  as  in  multiplication  of  simple  numbers. 
The  product  mil  be  in  the  lowest  denomination  in  the  question,  which 
must  he  pointed  off  as  in  reduction  of  United  States  money.     (Art.  71.) 

Proof.  —  The  proof  is  the  same  as  in  multiplication  of  simple 
numbers. 

EXAJEPLES   FOR   PRACTICE. 

1.  What   will   143   barrels  2.  What  will    144   gallons 

of   flour   cost   at    $  7.25   per         of  oil  cost  at  $  1.625  a  gal- 
barrel  ?        Ans.  $  1036.75.  Ion  ?  Ans.  $  234. 

OPERATION.  OPERATION. 

Multiphcand  $  7.2  5  MultipUcand       $  1.6  2  5 

Multiplier  14  3  Multiplier  14  4 

2175  6500 

2900  6500 

725  1  625 


Product  $  1  0  3  6.7  5  Product         $  2  3  4.0  0  0 

3.  What  will  165  gallons  of  molasses  cost  at  $  0.27  a  gal- 
lon ?  Ans.  $  44.55. 

4.  Sold  73  tons  of  timber  at  $  5.68  a  ton ;  what  was  the 
amount?  Ans.  $414.64. 

5.  What  will  43  rakes  cost  at  $  0.17  apiece  ?       Ans.  $  7.31. 

6.  What  will  19  bushels  of  salt  cost  at  $  1.625per  bushel  ? 

Ans.  $  30.875. 

7.  What  will  47  acres  of  land  cost  at  $  37.75  per  acre  ? 

Ans.  $  1774.25. 

8.  What  will  19  dozen  penknives  cost  at  $0,375  apiece? 

Ans.  $  85.50. 

9.  What  is  the  value  of  17  chests  of  souchong  tea,  each  weigh- 
ing 59  pounds,  at  $  0.67  per  pound  ?  Ans.  $  672.01. 

10.  When  19  cords  of  wood  are  sold  at  $  5.63  per  cord, 
what  is  the  amount?  Ans.  $  106.97. 

11.  A  merchant  sold  18  barrels  of  pork,  each  weighing  200 
pounds,  at  12  cents  5  mills  a  pound ;  what  did  he  receive  ? 

Ans.  $  450. 

»■■■■■   ■        -  — ^  I    —    -I „  ■         .  -■  ■  '         ■  ■■■—..  ^ 

74.  IIow  do  you  arrange  the  multiplicand  and  multiplier  in  multiplication 
of  United  States  money  ?  How  multiply  ?  Of  wliat  denoniiuation  is  tho 
product  1    How  must  it  be  pointed  off  1    Repeat  the  rule. 


DIVISION.  75 

12.  What  cost  132  tons  of  hay  at  $  12.125  per  ton  ? 

Ans.  $1G00.50. 

13.  A  fai-iner  sold  one  lot  of  land,  containing  187  acres,  at 
$  37.50  per  acre  ;  another  lot,  containing  89  acres,  at  $  137.37 
per  acre  ;  and  another  lot,  containing  57  acres,  at  $  89.29  per 
acre ;  what  was  the  amount  received  ibr  the  whole  ? 

Ans.  Z  24327.96. 
DIVISION. 

75i     Rule. — Divide  as  in  division  of  simpU  numbers. 
The  quotient  will  he  in  the  lowest  denomination  of  the  dividend,  which 
must  he  pointed  off  as  in  reduction  of  United  Slates  money.     (Art.  71.) 

Note.  —  When  the  dividend  consists  of  dollars  only,  and  is  either  smaller 
than  the  divisor  or  cannot  be  divided  by  it  without  a  remainder,  reduce  it  to 
a  lower  denomination  by  annexing  two  or  three  ciphers,  as  the  case  may 
requke,  and  the  quotient  will  be  cents  or  mills  accordingly. 

Proof.  —  The  proof  is  the  same  as  in  division  of  simple 
numbers. 

Examples  for  Practice. 

1.  If  59  yards  of  cloth  cost  2.  Purchased  68  ounces  of 

$  90.27,  what  will  1  yard  cost?      indigo  for  $17.     What  did  I 
Ans.  $  1.53.  give  per  ounce  ? 

Ans.  $  0.25. 

OPERATION.  OPERATION. 

Dividend.      $.  Dividend.      $. 

Divisor  5  9  )  9  0.2  7  (  1.5  3  Quotient.    Divisor  6  8  )  1  7.0  0  (  0.2  5  Quotient 

59  136 


312  340 

295  340 


177 
177 


3.  If  89  acres  of  land  cost  $  12225.93,  what  is  the  value  of 
1  acre  ?  Ans.  $  137.37. 

4.  When  19  yards  of  cloth  are  sold  for  $  106.97,  what  should 
be  paid  for  1  yard  ?  Ans.  $  5.63. 


75.  How  do  you  airange  the  dividend  and  divisor  in  division  of  United 
States  money?  How  divide?  Of  what  denomination  is  the  quotient?  How 
pointed  off?  How  do  you  proceed  when  the  dividend  is  dollars  only,  and  is 
either  smaller  than  the  divisor  or  cannot  be  divided  by  it  without  a  remain- 
der ?    The  rule  * 


76  UNITED    STATES   M0:NEY. 

5.  Gave  $  22.50  for  18  barrels  of  apples  ;  what  was  paid  for 
1  barrel  ?     For  5  barrels  ?     For  10  barrels  ? 

Ans.  $  20  for  all. 

6.  Bought  153  pounds  of  tea  for  $  90.27  ;  what  was  it  per 
"pound  ?  Ans.  $  0.59. 

7.  A  merchant  purchased  a  bale  of  cloth,  containing  73  yards, 
for  $  414.64 ;  what  was  the  cost  of  1  yard  ?  Ans.  $  5.68. 

8.  If  126  pounds  of  butter  cost  $16.38,  what  will  1  pound 
cost?  .  Ans.  $0.13. 

9.  If  63  pounds  of  tea  cost  $  58.59,  what  will  1  pound  cost  ? 

Ans.  $0.93. 

10.  If  76  cwt.  of  beef  cost  $249.28,  what  wiU  1  cwt.  cost? 

Ans.  $3.28. 

11.  If  96,000  feet  of  boards  cost  $  1120.32,  what  will  a  thou- 
sand feet  cost  ?  Ans.  $11.67. 

12.  Sold  169  tons  of  timber  for  $  790.92  ;  what  was  received 
for  1  ton  ?  Ans.  $  4.68. 

13.  When  369  tons  of  potash  are  sold  for  $48910.95,  what  is 
received  for  1  ton  ?  Ans.  $  132.55. 

14.  For  19  cords  of  wood  I  paid  $  109.25  ;  what  was  paid  for 
1  cord  ?  Ans.  $  5.75. 

PRACTICAL   QUESTIONS  BY   ANALYSIS. 

76i  Analysis  is  an  examination  of  a  question  by  resolving 
it  into  its  parts,  in  order  to  consider  them  separately,  and  thus 
render  each  step  in  the  solution  plain  and  intelligible. 

77i  The  price  of  one  pound,  yard,  bushel,  &c.,  being 
given,  to  find  the.  price  of  any  quantity. 

Rule.  —  Multiply  the  price  by  the  quantity. 

Ex.  1.  K  1  ton  of  hay  cost  $  12,  what  will  29  tons  cost? 

Ans.  $  348. 

Illustration.  —  Since  1  ton  costs  $12,  29  tons  will  cost  29  times 
as  much :  $  12  X  29  =  S  348. 

2.  If  1  bushel  of  salt  cost  93  cents,  what  will  40  bushels  cost  ? 
What  will  97  bushels  cost  ?  Ans.  $  90.21. 


77.  The  price  of  1  pound,  &c.,  bcinj^  given,  how  do  you  find  the  price 
any  quantity  1     The  rcuson  for  this  rule  ? 


of 


ANALYSIS.  77 

3.  If  1  bushel  of  apples  cost  $  1.65,  what  will  5  bushels  cost  ? 
What  will  18  bushels  cost?  Ans.  $  29.70. 

4.  If  1  ton  of  clay  cost  $  0.67,  what  will  7  tons  cost  ?  AVhat 
will  63  tons  cost  ?  Ans.  $  42.21. 

5.  When  $  7.83  are  paid  for  1  cwt.  of  sugar,  what  will  12  cwt. 
cost  ?     What  will  93  cwt.  cost?  Ans.  $  728.19. 

6.  When  $  0.09  are  paid  for  1  lb.  of  beef,  what  will  12  lb. 
cost  ?     What  will  760  lb.  cost  ?  Ans.  $  68.40. 

7.  A  gentleman  paid  $  38.37  for  1  acre  of  land ;  what  was 
the  cost  of  20  acres  ?     What  would  144  acres  cost  ? 

Ans.  $5525.28. 

8.  Paid  $  6.83  for  1  barrel  of  flour ;  what  was  the  value  of 
9  barrels  ?     What  must  be  paid  for  108  bai-rels  ? 

Ans.  $737.64. 

78.  The  price  of  any  quantity  being  given,  to  find  the 
price  of  a  unit  of  that  quantity. 

Rule.  —  Divide  tlie  price  by  (lie  quantity. 

9.  If  15  bushels  of  corn  cost  $  10.35,  what  will  1  bushel  cost  ? 

Ans.  $0.69. 

Illustration.  —  Since  15  bushels  cost  S  10.35,  1  bushel  will  cost 
one  fifteenth  part  of  $  10.35  ;  and  one  fifteenth  of  $  10.35  =  $  10.35 
-^15  =.$0.69. 

10.  Bought  65  barrels  of  flour  for  $  422.50  ;  what  cost  one 
barrel  ?     What  cost  15  barrels  ?  Ans.  $  97.50. 

11.  For  45  acres  of  land  a  farmer  paid  $2025  ;  what  cost  1 
acre  ?     What  180  acres  ?  Ans.  $  8100. 

12.  For  5  pairs  of  gloves  a  lady  paid  $  3.45 ;  what  cost  1 
pair?     What  cost  11  pairs?  ■  Ans.  $7.59. 

13.  If  11  tons  of  hay  cost  $214.50,  what  will  1  ton  cost? 
What  will  87  tons  cost  ?  Ans.  $  1696.50. 

14.  When  $  60  are  paid  for  8  dozen  of  arithmetics,  what  will 
1  dozen  cost  ?     What  will  87  dozen  cost  ?  Ans.  $  652.50. 

15.  Gave  $  5.58  for  9  bushels  of  potatoes ;  what  will  1  bushel 
cost  ?     What  will  43  bushels  cost  ?       ,  Ans.  $  26.66. 

16.  Bought  5  tons  of  hay  for  $85  ;  whd,t  would  1  ton  cost? 
What  would  97  tons  cost?  Ans.  $  1649. 

78.  How  do  you  find  the  price  of  1  pound,  &c.,  the  price  of  any  quantity 
being  given  1     The  reason  for  this  rule  ? 

7* 


78  UNITED   STATES  MONEY. 

17.  If  J.  Ladd  will  seU  20  lb.  of  butter  for  $  3.80,  what 
should  he  charge  for  59  lb.?  Aus.  $  11.21. 

18.  Sold  27  acres  of  land  for  $  472.50  ;  what  was  the  price  of 
1  acre  ?     What  should  be  given  for  12  acres  ?         Ans.  $  210. 

19.  Paid  $  39.69  for  7  cords  of  wood ;  what  will  1  cord  cost? 
What  will  57  cords  cost  ?  Ans.  $  323.19. 

20.  Paid  $  10.08  for  144  lb.  of  pepper ;  what  was  the  price 
of  1  pound  ?     What  cost  359  lb.  ?  Ans.  $  25.13. 

21.  Paid  $  77.13  for  857  lb.  of  rice  ;  what  cost  1  lb.  ?  What 
cost  359  lb.  ?  Ans.  $  32.31. 

22.  J.  Johnson  paid  $  187.53  for  987  gal  of  molasses;  what 
cost  1  gal.  ?     What  cost  329  gal.  ?  Ans.  S  62.51. 

23.  For  47  bushels  of  salt  J.  Ingersoll  paid  S  26.32  ;  what 
cost  1  bushel  ?     What  cost  39  bushels  ?  Ans.  $  21.84. 

79.  The  price  of  any  quantity  and  the  price  of  a  unit 
of  that  quantity  being  given,  to  find  the  quantity. 

Rule.  —  Divide  the  whole  price  by  the  price  of  a  unit  of  the  quantity 
required. 

24.  If  I  expend  $  150  for  coal  at  $  6  per  ton,  how  many  tons 
can  I  pui'chase  ?  Ans.  25  tons. 

Illustration.  —  Since  I  pay  $  6  for  1  ton,  I  can  purchase  as  many- 
tons  with  S  150  as  $  6  is  contained  tunes  in  S  150  :  $  150  -^  S  6  =  25 ; 
therefore  I  can  purchase  25  tons. 

25.  At  S  5  per  ream,  how  many  reams  of  paper  can  be  bought 
for  $  175  ?  Ans.  35  reams. 

26.  At  $  7.50  per  barrel,  how  many  barrels  of  flour  can  be 
obtained  for  $  217.50  ?  Ans.  29  bai-rels. 

27.  At  $  75  per  ton,  how  many  tons  of  iron  can  be  purchased 
for  $  4875  ?  Ans.  65  tons. 

28.  At  $  4  per  yard,  how  many  yards  of  cloth  can  be  bouglif 
for  $  1728  ?  Ans.  432  yards. 

29.  IIow  many  hundred  weight  of  hay  can  be  bought  for 
$9.66,  if  $0.69  are  paid  for  1  hundred  weight? 

Ans.  14  hundred  weight. 

30.  If  $  66.51  are  paid  for  flour  at  $  7.39  per  barrel,  how 
many  barrels  can  be  bought  ?  Ans.  9  barrels. 

31.  Paid  $136.50  for  wood,  at  $3.25  per  cord;  how  many 
cords  did  I  buy  ?  Ans.  42  cords. 

79.  How  do  you  find  tlio  quantity,  the  price  of  1  pound,  &c.,  being  given  1 
The  reason  for  the  rule  1 


BILLS.  79 


BILLS. 


80.  A  Bill  is  a  paper,  given  by  merchants,  containing  a  state- 
ment of  goods  sold,  and  their  pi'ices. 

An  Invoice  is  a  bill  of  merchandise  shipped  or  forwarded  to  a 
purchaser,  or  selling  agent. 

The  Dale  of  a  bill  is  the  time  and  place  of  the  transaction. 

The  bill  is  against  the  party  owing,  and  in  favor  of  the  party 
who  is  to  receive  the  amount  due. 

A  bill  is  receipted  when  the  receiving  of  the  amount  due  is 
acknowledged  by  the  party  in  whose  favor  it  is.  A  clerk,  or  any 
other  authorized  person,  may,  in  his  stead,  receipt  for  him,  as  in 
bill  2. 

"Wlien  the  bill  is  in  the  form  of  an  account,  containing  items 
of  debt  and  credit  in  its  settlement,  it  is  requh'ed  to  lind  the 
difference  due,  or  balance,  as  in  bill  5. 

What  is  the  cost  of  each  article  in,  and  the  amount  due  of, 
each  of  the  following  bills  ? 

(1.)  New  York,  May  20,  1856. 

Dr.  John  Smith, 

Bougid  of  Somes  &  Gridlet, 

82  gals.  Temperance  Wine,        at        $  0.75 
89     "     Port  do.  "  .92 

2^  pairs  Silk  Gloves,  "  .50 


Received  payment, 


Ijp  loo. do. 


Somes  &  Gridley. 


(2.)  PJdladelphia,  March  7,  1857. 

Mr.  Levi  Webster, 

Bought  o/ James  Frankland, 

6  lbs.  Giocolate,  at  $  0.18 

12    "    Flour,  «  .20 

&  pairs  Shoes,  "  1.80 

30  lbs.  Candles,  «  .26 

$22.08. 

Received  payment, 

James  Frankland, 

hy  Enoch  Osgood. 

80.  What  is  a  bill  ?     What  is  an  invoice  ?     When  is  a  bill  against,  and 
when  in  favor,  of  a  party  ?     How  is  a  bill  receipted  1 


80 


UNITED   STATES   MONEY. 


(3.) 

Mr.  "William  Greenleaf, 

86  Shovels, 
90  Spades, 
18  Plows, 
23  Handsaws, 
14  Hammers, 
12  Mill-saws, 
46  cwt.  Jron, 


at 


St.  Louis,  March  19,  1856. 

Bought  o/"  Moses  Atwood. 

$0.50 

.86 

.       11.00 

3.50 

.62 

12.12 

12.00 

$1105.02. 


i 


Mr.  Amos  Dow, 


Boston,  June  5,  1856. 
Bought  of  Lord  &  Greenleaf, 


37  Chests  Green  Tea, 

42  "      Black   do. 

43  Casks  Wine, 

12  Crates  Liverpool  Ware, 
19  hbl.  Genesee  Flour, 
23  hu.  Rye, 


at 


$23.75 

17.50 

99.00 

175.00 

7.00 

152 


$8138.71 


Mr. 

1855. 
Apr.  5. 
Aug.  7. 

Nov.  1. 


1856. 
Jan.   1. 

Feb.  10. 

Apr.  3. 

u       a 


(5.) 

John  Wade, 


San  Francisco,  May  13,  1856. 
To  Ater,  Fitts,  &  Co.,  Dr. 


To  80  joo/rs  J?bse, 
"17     "      Boots, 
"19     "      Shoes, 
«   23     "      G^/(we5, 


at 
« 


$1.20 

3.00 

1.08 

.75 


$  184.77 


Or. 
27  Young  Headers, 
10  Greek  Lexicons, 
7  Webster's  LHctionaries, 

19  Folio  Bibles, 

20  Testaments, 


at 

(( 
(I 
u 


$0.20 

3.90 

4.75 

2.93 

.37 


Balance  due  A.,  F.,  &  Co. 
Received  payment. 


$140.72. 


$  44.05. 
Ayer,  Fitts,  &  Co. 


\ 


LEDGER  ACCOUNTS.  81 

LEDGER  ACCOUNTS. 

81.  A  ledger  is  the  principal  book  of  accounts  among  mer- 
chants. In  it  are  brought  together  scattered  items  of  account, 
olten  making  long  columns.  As  a  rapid  way  of  finding  the 
amount  of  each,  accountants  generally  add  more  than  one  column 
at  a  single  operation.     (Art.  24.) 

The  examples  below  may  be  added  both  by  the  usual  method 
and  by  that  which  is  more  rapid. 


1. 

2. 

3. 

4. 

$  cts. 

$  cts. 

$  cts. 

$   cts. 

5.7  5 

1.0  5 

7  1.10 

1  0  0.8  8 

3.15 

7.0  8 

3  5.6  0 

3  2  0.1  2 

6.3  7 

6.3  8 

2  1.4  0 

2  8  0.4  7 

10.13 

5.5  0 

1  0  0.5  0 

1  5  1.5  3 

5.0  5 

3.2  5 

6  2.7  5 

•   .9  2 

12.5  0 

8.19 

13.13 

1  1.0  8 

8.0  0 

1.13 

1.3  7 

4913 

.6  3 

10.10 

1  6.0  2 

4  4.2  2 

1.3  7 

15.2  5 

19.2  8 

6  0.8  1 

2  2.0  0 

13.4  5 

1  6  3.3  5 

5  2.7  5 

1  6.0  5 

6.17 

6  2  0.5  0 

3  5.15 

1.19 

.0  9 

7  5.0  0 

7  0.0  6 

.3  1 

1.13 

2  5.2  0 

105  0.0  0 

1  0.0  0 

8.0  7 

5  3.8  1 

3  120.12 

1  1.8  8 

1  1.0  6 

3  3.1  9 

2  0  0.5  0 

.12 

3  5.15 

17.0  0 

16.0  9 

9.17 

1  8.9  1 

10.3  8 

9  0  0.11 

.3  3 

1  0.0  3 

4  0.12 

1  8  2  5.5  0 

6.2  2 

3  0.0  0 

15.6  8 

10  5.10 

2.3  1 

1.8  8 

7  1.12 

3  5.4  6 

7.17 

2.7  5 

13.19 

6  7.6  3 

1  5.5  0 

1.2  5 

1  0.0  0 

8  1.17 

1  1.2  5 

5.0  0 

18.2  0 

10.14 

.0  9 

2  5.5  0 

13.15 

7  5.0  0 

2  1.1  7 

1  2.0  2 

2  5.0  0 

1  2  0.0  0 

3  2.0  0 

1  9.1  7 

]  0  2.5  5 

1  1  4.0  9 

14.0  6 

3  2.4  3 

e^i  1.1 0 

2  12.6  3 

2  0.5  0 

4  6.3  7 

2  3  5.8  3 

1  0  3  0  0.4  8 

81 .  What  is  a  ledger  ?     How  may  ledger  columns  be  added  rapidly  1 


82 


REDUCTION. 


REDUCTION. 

82.  A  Simple  Nnmber  is  a  unit  or  a  collection  of  units,  either 
abstract,  or  concrete  of  a  single  denomination ;  thus,  1  dollar, 
9  apples,  12,  are  simple  numbers. 

A  Compound  Number  is  a  collection  of  concrete  units  of  several 
denominations,  taken  collectively ;  thus,  12£  18s.  9d.  is  a  com- 
pound number. 

A  Denominate  Number  is  any  concrete  number  which  may  be 
changed  to  a  different  denomination. 

Note.  —  A  Scale  denotes  the  relations  between  the  different  units  of  a 
number.  Simple  numbers  have  a  uniform  scale  of  10,  but  compoimd  numbers 
generally  have  a  varying  scale. 

83.  Reduction  is  the  changing  of  a  number  into  one  of  a  dif- 
ferent denommation,  but  of  equal  value. 

It  is  of  two  kinds,  Reduction  Descending  and  Reduction  As- 
cending. 

Reduction  Descending  is  the  changing  of  a  number  of  a  higher 
denomination  into  one  of  a  lower  denomination ;  as  pounds  to 
shillings,  &c.     It  is  performed  by  multiphcation. 

Reduction  Ascending  is  the  changing  of  a  number  of  a  lower 
denomination  into  one  of  a  higher  denomination ;  as  farthings 
to  pence,  &c.  It  is  the  reverse  of  Reduction  Descending,'  and 
is  performed  by  division. 

ENGLISH  MONEY. 

84.  English  or  Sterling  Money  is  the  Currency  of  England. 


4  Farthings  (qr.  or  far.) 

12  Pence 

20  Shillings 

21  Shillings 
20  Shillings 


£. 
1 


1 

20 


TABLE, 
make 


1  Penny, 

d. 

1   Shilling, 

8. 

1  Pound, 

£. 

1  Guinea, 

G. 

1  Sovereign, 

sov, 

<L 

for. 

1              = 

4 

12             = 

48 

240             = 

960 

82.  What  is  a  simple  number?     A  compound  number? — 83.  What  is 

reduction?     How  many  kinds  of  reduction  ?     What  are  they?  What  is 

reduction    dcsccndinj;  ?      Reduction    ascending  ?  —  84.  Wiiat    is  English 
money  ?     Repeat  the  table. 


REDUCTION.  83 

Note  1.  —  £  stands  for  the  Latin  -word  liln-a,  signifying  a  pound  ;  s.  for 
solii/us,  a  shilliug ;  d.  for  denarius,  a  penny ;  qr.  for  qaadrans,  a  quarter. 

Note  2.  —  Farthings  arc  sometimes  expressed  in  a  fraction  of  a  penny ; 
thus,  1  far.  =  ^  d.  ;  2  far.  =  ^  d. ;  3  far.  =  |  d. 

Note  3.  —  Tlie  Pound  Sterling  is  repre.scntcd  by  a  gold  coin  called  a  sov- 
ereign.    Its  usual  current  value  in  United  States  money  is  $4.84. 

Note  4.  —  The  term  sterling  is  proI)ahly  from  Easterling,  the  popular  name 

of  certain  early  German  traders  in  England. 

* 

Mental  Exercises. 

1.  How  many  farthings  in  3  pence  ?     In  9  pence? 

2.  How  many  pence  in  2  shillings  ?     Li  6  shilHngs  ? 

3.  How  many  shilhngs  in  7  pounds  ?     In  10  pounds  ? 

4.  How  many  pence  in  8  farthings  ?     In  24  farthings  ? 

5.  How  many  shillings  in  24  pence  ?     In  60  pence  ? 

6.  How  many  pounds  in  40  shilhngs  ?     In  80  shilhngs  ? 

Exercises  for  the  Slate. 

85.   To  reduce  from  a  higher  to  a  lower  denomination. 
Ex.  1.  How  many  farthings  in  17£  8s.  9d.  3far.  ? 

OrERATION. 

1  7£  8s.  9d.  3far.  We  multiply  the  17  by  20,  because 

.20  ^^  shillings  make  1  pound,  and  to  this 

■ _    _  product  we  add  the   8  shilhngs.     We 

3  48  shillings.  then  multiply  by  12,  because  12  pence 

1  2  make    1    shilling,  and   to   the   product 

A-ior  '"'^  '^^'l  tb^   ^^-     Again,  we  multiply 

4  i  »  o  pence.  i^^  4^  because  4  farthings  make  1  pen- 

4  ny,  and  to  this  product  we   add   the 


Ans.  16  7  4  3  farthings.        ^  ^^•'  ^""^  ^^^^^^^  ^'^^^^  farthings. 

Rule.  —  Multiply  the  highest  denomination  given  by  the  number  re- 
quired of  the  next  lower  denomination  to  make  one  in  the  denomination 
multiplied.  To  this  product  add  the  corresponding  denomination  of  the 
multiplicand,  if  there  be  any.  Proceed  in  like  manner,  till  the  reduction 
is  brought  to  the  denomination  required. 

85.  How  do  you  reduce  pounds  to  shillings  ?  Why  multiply  hy  20  ?  How 
do  you  redune"  shillings  to  pence?  Whv  ?  Pence'  to  farthings?  Why? 
Guineas  to  shilhngs  1     The  general  rule  for  reduction  descending  ? 


84  REDUCTION. 

86.  To  reduce  from  a  lower  to  a  higher  denomination. 
Ex.  2.  How  many  pounds  in  16743  farthings  ? 

OPERATION.  We  divide  by  4,  because  4  farthings 

4)16743  far.  make   1  peimy,  and  the  result  is  41S5 

pence,   and  the   remainder,    3,  is   far- 

1  2  )  41  8  5  d.  3far.  things.     We  divide  by  12,  because  12 

20^348'?   9d  pence  make  1  shilling,  and  the  result  is 

^ ""'       '  348  shillings,   and  the   9   remaining  is 

1  7  £  83.  pence.     Lastly,  -we  divide   by  20,  be- 

Ans.  17£  8s.  9d.  3far.  ^f"^^  ^O  shillings  make  1  pound,  and 

the  result  is  Itk,  8s.;  and  by  annexing 
all  the  remainders  to  the  last  quotient,  we  find  the  answer  to  be  1 7£ 
8s.  9d.  3far. 

B,ULE.  —  Divide  the  lowest  given  denomination  hy  the  number  which  it 
takes  of  that  ctenomination  to  make  one  of  the  denomination  next' higher. 
The  quotient  thus  obtained  divide  in  like  manner,  and  so  proceed  until 
ft  is  brought  to  the  denomination  required.  The  last  quotient,  with  the 
remainders  connected,  will  be  the  answer. 

3.  In  9£  I83.  7d.  how  many  pence  ? 

4.  In  2383d.  how  many  pounds,  &c.  ? 

5.  How  many  farthings  in  14£  lis.  5d.  2far.  ? 

6.  How  many  pounds  iu  13990far.  ? 

TROY  WEIGHT. 

87.  Troy  Weight  is  the  weight  used  in  weigliing  gold,  silver,  and 
jewels. 

TABLE. 


24  Grains  (gi 

•) 

make 

1  Pennyweight, 

pwt. 

20  Pennj-wei 
12  Ounces 

^hts 

(1 

1  Ounce, 
1  Pound, 

oz. 

lb. 

lb.           = 
1            = 

oz. 

1 

12 

= 

pwt. 

1 

20 
240 

= 

24 

480 
5760 

86.  How  do  you  rnhu-c  f;ntliin£rs  to  ponco  ■»  AVIiy  divide  hy  4  ?  TTow 
do  von  reduce  fx^nro  to  sliilliiiL's  ?  WIiv  ?  Sliillinsrs  to  pounds  ?  Why  1 
Shiilines  to  puifion's  ?  Wluit  is  the  <roncrMl  vide  for  i-eduction  ascending? 
Wliat  is  Troy  Weiglit  used  for  ?    Repeat  tlic  table. 


REDUCTION.  -  85 

Note  1. —  Thj  oz.  stands  for  onza,  the  Spanish  for  ounce. 

Note  2.  —  A  grain  or  corn  of  wheat,  gatiiered  out  of  the  middle  of  the 
ear,  was  the  origin  of  all  the  weights  used  in  England.  Of  these  grains,  32, 
well  dried,  were  to  make  one  pcnnywciglit.  But  in  later  times  it  was  thought 
sufficient  to  divide  the  same  pennyweight  into  24  equal  parts,  still  called 
grains. 

Note  3.  —  Diamonds  and  other  precious  stones  are  weighed  by  what  is 
called' Z>/amo«a?  Weight,  of  which  16  parts  make  1  grain;  4  grains,  1  carat. 
1  grain  Diamond  Weight  is  equal  to  f  of  a  grain  Troy. 

Note  4.  —  The  Troy  pound  is  the  standard  unit  of  weight  adopted  by  the 
United  States  Mint,  and  is  the  same  as  the  Imperial  Troy  pound  of  Great 
Britain. 

Men^^l  Exercises. 

1.  How  many  gr.  in  2pwt.  ?     In  lOpwt.  ? 

2.  How  many  pennyweights  in  4oz.  ?     In  20oz.  ? 

3.  How  many  ounces  in  21b.  ?     In  51b.  ?     In  101b.  ? 

4.  How  many  pennyweights  in  48gT.  ?     In  96gr.  ? 

5.  How  many  ounces  in  40pwt.  ?     In  120pwt.  ? 

6.  How  many  pounds  in  24oz.  ?     In  60oz  ?     In  120oz.  ?  , 

Exercises  for  the  Slate. 

1.  How  many  grains  in  721b.  2.  In    419887   grains,   how 

lOoz.  15pwt.  7gr.  ?  many  pounds  ? 

OPERATION.  OPERATION. 

7  2  lb.  lOoz.  15pwt.  7gr.     2  4  )  4  1  9  8  8  7  gr. 

^  ^  2  0)17495  pwt.  7gr. 


8  7  4  ounces.  -\  o  \  q  n  a         -ik      ^ 

o/^  12)874  oz.  15pwt. 

7  2  lb.  20oz. 


17  4  9  5  pennyweights. 

24  Ans.  721b.  lOoz.  15pwt.  7gr. 

69987 
34990 


Ans.  419  8  8  7  grains. 


87.  What  was  the  original  of  all  weights  in  England  ?  How  many  of  these 
gi-ains  did  it  take  to  makj  a  pennyweight  1  How  many  grains  in  a  penny- 
weight now  ?  By  what  weight  are  diamonds  weighed  ?  What  is  the  stand- 
ard at  the  mint  ?  How  do  you  reduce  pounds  to  grains  ?  Gi-ve  the  reasoa 
of  the  operation.  How  do  you  reduce  grains  to  pounds  ? 
8 


86  .  REDUCTION. 

3.  How  many  grains  in  76pwt.  12gr.  ? 

4.  How  many  pennyweights  in  1836gr.  ? 

5.  In  761b.  5oz.  how  many  grains  ? 

6.  In  440160  grains  how  many  pounds  ? 

7.  How  many  pennyweights  in  1441b.  9oz.  ? 

8.  How  many  pounds  in  34740pwt.  ? 
•    9.  How  many  pounds  in  17895gr.  ? 

10.  In  31b.  loz.  opwt.  logr.  how  many  grains? 

11.  A  valuable  gem  weighing  2oz.  18pwt.  12gr.  was  sold  for 
$  1.37  per  grain  ;  what  was  the  sum  paid  ?       Abs.  $  1923.48. 

APOTHECARIES'    WEIGHT. 
88.  Apothecaries'  Weight  is  used  in  mixing  medicines. 


TABLE. 

20  Grains  (^ 

?r.) 

make           1  Scruple, 

sc.  or  3 

3  Scruples 

"                1  Dram, 

dr.  or  3 

'  8  Drams 

"               1  Ounce, 

oz.  or  S 

12  Ounces 

«               1  Pound, 
sc. 

lb.  or  ft. 

dr.                            1 

= 

20 

oz. 

1         =             3 

= 

60 

lb 

1 

=           8         =           24 

= 

480 

1         = 

12 

=         96         =         288 

= 

5760 

Note  1  —  In  this  weight  the  pound,  ounce,  and  grain  are  the  same  as  in 
Troy  Weight. 

Note  2.  —  Medicines  are  usually  bought  and  sold  by  Avoirdupois  "Weight. 

Note  3.  —  Of  fluids,  4.5  drops,  or  a  common  tea-spoonful,  make  about  1 
fluid  dram  ;  2  common  table-spoonfuls,  about  1  fluid  ounce. 

Mental  Exercises. 

1.  In  40  grams  how  many  scru]des  ?     In  60gr.  ?     In  120gr.  ? 

2.  In  5  scruples  how  many  grains  ?     In  lOsc.  ?     In  40sc.  ? 

3.  In  3  drams  how  many  scruples  ?     In  lOdr.  ?     In  17dr.  ? 

4.  How  many  pounds  in  48  ounces?     In  96oz.  ?     In  144oz.  ? 

5.  How  many  ounces  in  24  drams  ?     In  64dr.  ?     In  96dr.  ? 

88.  For  what  is  Apothopario"'  Wcisrht  used  ?  Whnt  d<^nominntions  of  this 
weight  are  the  same  as  tho-i'  of  Troy  W<'i<.rht  ?  By  wliat  weight  arc  medi- 
cinea  usually  bought  and  sold  1     Repeat  the  table. 


REDUCTION.  87 

f 

Exercises  for  the  Slate. 

1.  In  401b.  8oz.  5dr.  Isc.  7gr.  2.    How    many   pounds    in 

how  many  grains  ?  234567  grains  ? 

OPERATION.  OPERATION. 

4  0  lb.  8oz.  5dr.  Isc.  7gr.    2  0)234  5^  gr. 
12 


3  )  1  1  7  2  8  sc.  7gr. 
4  8  8  i^unces.  8)3909  dr.  Isc. 

1  2  )  4  8  8  oz.  5dr. 
3  4  0  lb.  8oz. 


3  9  0  9  drams. 


117  2  8  scruples. 

2  0,  Ans.  401b.  8oz.  5 dr.  Isc.  7gr. 

Ans.  2  3  4  5  6  7  grains. 

3.  How  many  scruples  in  761b.? 

4.  How  many  pounds  in  218889  ? 

5.  How  many  grains  in  1441b.  ? 

6.  How  many  pounds  in  829440gr.  ? 

7.  In  121b  8§  3  3  19  18gr.  how  many  grains? 

8.  In  73178  grains  how  many  pounds  ? 

9.  In  7S  63  29  of  tartar  emetic,  how  many  doses  of  20gr. 
each?  ..  Ans.  188. 

AVOIRDUPOIS   TV'EIGHT. 

89i  Avoirdupois  Weight  is  used  in  weighing  almost  every  kind 
of  goods,  and  all  metals  except  gold  and  silver. 

TABLE. 


16  Drams  (dr.) 

make 

1  Ounce, 

oz. 

16  Ounces 

t( 

1  Pound, 

lb. 

25  Pounds 

(( 

1  Quarter, 

qr. 

4  Quarters 

(( 

1  Hundred  We 

ight, 

cwt. 

20  Hundred  Weight 

« 

1  Ton, 

oz. 

T. 

dr. 

lb.           ^   1 

== 

16 

qr. 

1       =            16 

= 

256 

cwt.                    1 

= 

25       =         400 

= 

6400 

T-                   1=4 

= 

100       =       1600 

= 

25600 

1       =       20       =       80 

= 

2000       =     32000 

= 

512000 

88  How  do  you  i-educe  pounds  to  p:rains "?  The  reason  for  the  operation  ? 
How  do  you  reduce  frrains  to  pounds  ?  The  reason  for  the  operation  ?  — 
89.  For  what  is  Avoirdupois  "Weight  used  ?     Recite  the  table. 


88 


REDUCTION. 


Note  1.  —  In  civt.  the  c  stands  for  centum,  the  Latin  for  one  hundred,  and 
wt.  for  weight. 

Note  2.  —  The  laws  of  most  of  the  States,  and  common  practice  at  the 
present  time,  make  ^5  pounds  a  quarter.  But  formerly,  28  pounds  were  al- 
lowed to  malce  a  quarter,  112  pounds  a  hundred,  and  2240  pounds  a  ton,  as 
is  still  the  standard  of  the  United  States  governnieut  at  the  custom-houses. 

Note  3.  — Avoirdupois  is  from  the  French  avoir  du  poid,  to  have  weight. 

Note  4.  —  1  pound  Avoirdupois  =  7000  gr.  Troy  =  lib.  2oz.  11  pwt.  16 
gr.  Troy;  lib.  Troy,  or  Apothecary  =  5760fir.  Troy  =  13oz.  2\\^  dr. 
Avoirdupois  ;  loz.  Troy,  or  Apoth.  =  480gr.  Troy  =  loz.  1^®^  dr.  Av. ; 
loz.  Av.  =  437^gr.  Troy  =  18pwt.  .5igr.  Troy ;  Idr.  Apoth.  =  60gr.  Troy  = 
2^^*^dT.  Av. ;  Idr,  Av.  =  27^gr.  Troy  =  Ipwt.  Sf^gr.  Troy  ;  Ipwt.  Troy  = 
24gr.  Troy  =  |-f|  of  a  dr.  Av. ;  Isc.  Apoth.  =  20  gr.  Troy  =  ^f|-  of  a  dr.  Av. 

Mental  Exercises. 

1.  How  many  drams  in  3oz.  ?    In  7oz.  ?    In  lOoz.  ?    In  12oz.  ? 

2.  How  many  ounces  in  101b.?    In  151b.?    In  121b.?   In  1001b.? 

3.  How  many  pounds  in  2  quarters  ?     In  3qr.  ?     In  20qr.  ? 

4.  How  many  quarters  in  lOcwt.  ?     In  IGcwt.  ?     Li  ITcAVt.  ? 

5.  How  many  tons  in  80cwt.  ?     In  lOOcwt.  ?     In  GOOcwt.  ?  • 

6.  How  many  hundred  weight  in  16qr.  ?    In  48(|r.  ?  In  96qr.  ? 


Exercises  for  the  Slate. 


1.  How  many  pounds  in  176T. 
17cwt.  3qr.  151b.  ? 

operation. 
1  7  6  T.  17cwt.  Sqr.  151b. 
20 


3  5  3  7  hundred  weight. 
4 


1415  1  quarters. 
25 


70770 
28302 


3  5  3  7  9  0  pounds,  Ans. 


how 


2.   In     3537901b. 
many  tons  ? 

operation. 
25)353790  lb. 

4  )  1  4  1  5^^  qr.  151b. 

2  0)3537  cwt.  3qr. 

1  7  6  T.  17cwt 

176T.  17cwt.  3qr.  151b.  Ans. 


89.  How  many  pounds  are  now  allowed  for  a  cwt.,  and  how  many  for  a  quar- 
ter of  a  cwt.,  in  most  of  the  United  States,  in  I)uying  and  sellinir  articles  l)y 
weight  f  How  many  at  the  custom-housos  ?  How  do  you  nducc  tons  to 
drams  ?  The  reason  for  the  operation  1  How  do  you  reduce  drams  to  tons? 
The  reason  for  the  operation  1 


REDUCTION.  89 

3.  In  16T.  19cwt.  Oqr.  lOlb.  lloz.  5dr.  how  many  drams  ? 

4.  In  8681141  drams  how  many  tons? 

5.  In  679cwt.  how  many  pounds? 

6.  In  679001b.  how  many  cwt.  ? 

7.  "What   cost    17cwt.    oqr.    181b.    of  beef,    at    7    cents   per 
pound?  Ans.  $125.51. 

8.  What  cost  48T.  17cwt.  of  lead  at  8  cents  per  pound  ? 

•  Ans.  $7816.00. 

CLOTH    MEASURE. 

90.    Cloth  Measure  is  used  in  measuring  cloth,  ribbons,  lace,  and 
other  articles  sold  by  the  yard  or  ell. 

TABLE. 

2J  Inches  (in.)  make  1  Nail,  na. 

4    Nails  "  1  Quarter  of  a  Yard,  qr. 

4  Quarters  "  1  Yard,  yd. 

3    Quarters  «  1  Ell  Flemish,  E.  F. 

5  Quarters  «  1  Ell  Enghsh,  E.  E. 


qr.  1=2^ 

E.  F.  1        _         4        ==         9 

yd.  1  ==  3        =        12        =       27 

E.E.  '  1  ^  11  =  4        ^        16        =       36 

1  =  1^    ,     =:  1|-         =  5        =        20        =       45 

Note.  —  The  Ell  French  is  6  quarters  ;  the  Ell  Scotch,  4qr.  l^in. 

*  Mental  Exercises. 

1.  In  2  quarters  how  many  nails?     In  5qr.  ?     In  Sqr.  ?     In 
20qr.  ?     In  25qr.  ?     In  30qr.  ?     In  40qr.  ? 

2.  In  3  yards  how   many  quarters  ?      In  7yd.  ?      In  8yd.  ? 
In  14yd.?     In  19yd.?     In  100yd.  ?     In  200yd.? 

3.  How  many  quarters  in  8  nails  ?     In  20na.  ?     In  48na.  ? 

4.  How  many  yards  in  20  quarters  ?     In  40qr.  ?     In  lOOqr,  ? 

Exercises  FOK  the  Slate. 

1.  How  many  nails  in  47  yd.  2.  In  765  nails  how  many 

3qr.  Ina.  ?  yai'ds  ? 

90.  For  what  is  cloth  measure  used  ?     Repeat  the  table.    Is  the  ell  French 
longer  or  shorter  than  the  ell  English  1     What  makes  an  ell  Scotch  ? 
8* 


90 


REDUCTION. 


OPERATION. 


4  7  yd.  3qr.  Ina. 
4 


19  1  quarters. 
4 

Ans.  7  6  5  nails. 


OPERATION. 

4)7  65  na. 
4)191  qr.  Ina. 
Ans.  4  7  yd.  3qr.  Ina. 


3.  In  144yd.  3qr.  how  many  quarters  ? 

4.  In  579  quarters  how  many  yards  ? 

5.  In  17E.  E.  4qr.  3na.  how  many  nails  ? 

6.  In  359  nails  how  many  ells  English  ? 

7.  In  126yd.  Oqr.  3na.  how  many  nails? 

8.  In  2019  nails  how  many  yards? 

9.  What  cost  49yd.  3qr.  of  cloth,  at  $  2.17  per  quarter  ? 

Ans.  $431.83. 

10.  What  cost  144yd.  Iqr.  3na.  of  cloth,  at  25  cents  per  luul? 

Ans.  $  577.75. 

LONG  MEASURE. 
91.   linear  or  long  Measure  is  used  in  measuring  distances  in 


any  direction. 

TABLE. 

12    Inches  (in.)                ma 
3    Feet                                ' 
51  Yards,  or  16^  Feet       ' 

40    Rods                               • 
8    Furlongs,  or  320  Rods,  ' 
3    Miles                              ' 

691  ]\lilos  (nearly)                ' 
360    Degrees                          ' 

ke 

Foot, 

Yard, 

Rod,  or  Pole, 

Furlong, 

Mile, 

League,  .^ 

Degree, 

Circle  of  the  Earth. 

d( 

ft. 

rd. 
fur. 

m. 
lea. 

^g.  or  °. 

a 

in. 

rd. 

fur.                        1 

m.                    1          ^        40 

1       =       8         =320 

= 

yd. 

1       = 

220       = 
1760       = 

1 

3 

660 
5280 

= 

12 

3G 

198 

7920 

63360 

90.  How  do  you  reduce  yards  to  nails  ?  How  do  you  reduce  nails  to  yards? 
The  reason  for  the  operation  '.  —  91.  How  is  linear  or  long  measure  used  ? 
Kcpeat  tiic  table. 


EEDUCTION.  91 

Note  1.  —  12  lines  make  1  inch  ;  4  inches,  1  hand  ;  6  feet,  1  fathom  ;  -g^o 
of  a  degree  of  the  circumference  of  the  earth,  1  knot,  or  geographical  miie> 
equal  to  l\^  statute  miles. 

Note  2.  —  The  yai-d  is  the  standard  unit  of  linear  measure  adopted  by  the 
United  States  government,  and  it  is  the  same  as  the  imperial  yard  of  Great 
Britain.  A  metre,  the  unit  of  linear  measure,  as  established  by  the  French 
government,  is  equal  to  about  39^^^  English  inches. 

Note  3.  —  The  English  statute  mile  is  the  same  as  that  of  the  United 
States,  but  that  of  other  countries  ditiers  in  value  from  it ;  as  the  German 
short  mile  is  equal  to  6857  yards,  or  about  Sjq  English  miles  ;  the  German 
long  mile,  to  10125  yards,  or  about  5|  English  miles;  the  Prussian  mile,  to 
8237  yards,  or  about  4^-^  English  miles  ;  the  Spanish  common  league,  to 
7416  yards,  or  about  4-^  English  miles ;  the  Spanish  judicial  league,  to  4635 
yards,  or  about  2f  English  miles. 

Note  4.  —  A  degree  of  longitude  is  yq-q  of  any  circle  of  latitude.  As 
the  circles  of  latitude  diminish  in  length,  the  degrees  of  longitude  vary  in 
length  under  different  parallels  of  latitude.  Thus,  under  the  equator,  the 
length  of  a  degree  of  longitude  is  about  69  J-  statute  miles;  at  25°  of  latitude, 
62iV  miles ;  at  40°  of  latitude,  53  miles  ;  at  42°  of  latitude,  51^  miles;  at 
49°  of  latitude,  45^  miles  ;  at  60°,  34^'^2  miles. 

Mental  Exercises. 

1.  How  many  inches  in  4  feet?  In  10ft.?  In  12fl.  ?  In 
20fL  ? 

2.  How  many  feet  in  two  yai-ds  ?  In  Syd.  ?  In  20y(I.  ?  In 
18yd.  ? 

3.  How  many  rods  in  2  furlongs  ?  In  8fur.  ?  In  Ifur.  ?  In 
30fur.  ?     In  lOOfur.  ?     In  200fur.  ?     In  400fur.  ? 

4.  How  many  leagues  in  9  miles  ?  In  21m.  ?  In  81m  ?  In 
144m.  ?     In  40m.  ?     In  50m.  ?     In  80m.  ? 

5.  How  many  furlongs  in  120  rods  ?    In  3G0rd.?    In  1440rd.  ? 

6.  How  many  yards  in  99  feet?     In  66ft.  ?     In  144ft.  ? 

7.  How  many  feet  in  108  inches  ?     In  144in.  ?     In  1728  in.  ? 

Exercises  for  the  Slate. 
1.  In  66deg.  56m.  7fur.  37rd.  12ft.  9in.  how  many  inches? 


91 .  How  many  lines  make  1  inch  ?  How  manv  inches  1  hand  ?  How  many 
feet  1  fathom  1  What  is  the  standard  unit  of  linear  measure  adopted  bv  the 
United  States  ?  Is  the  value  of  the  mile  the  same  in  all  countries?  How 
much  is  a  degree  of  longitude  under  the  equator  ?  At  40°  of  latitude  1  At 
42°  of  latitude  1    At  60°  of  latitude  ? 


92  DEDUCTION. 

OPERATION. 

6  6  deg.  o6m.  7fur.  37rd.  12ft.  9in. 
69^ 


2.  In   292849479  inches  how 


"00  many  degrees  ? 

401 

\   1  OPERATION. 

12)292849479 


4  6  2  1  miles. 

8  1  6  I  )24404123  ft.  Sin. 

3  6  9  7  5  fur.  _2     2 

, 11  3  3     )48808246     Ll2ft.  6in. 

14  7  9  0  3^  rods.  ^  q  )  1479037  rd.  25^2=; 

1479038  69^)4621  miles  7fur. 

739518A  6  6 


24404122^  ft:  415  )27726 

—  G  6  deg.   336  »f- 

2  9  2849  47  9in.Ans.  [6  =  56m. 

BBdeg.  56m.  7fur.  37rd.  12J^ft.  3in. 

A  =  6in. 


66         56      7       37      12       9Ans. 


Note.  —  To  multi])ly  by  ^,  we  take  ^  of  the  multiplicand.  —  To  divide 
by  16^,  we  first  reduce  both  the  divisor  and  dividend  to  halves,  and  then 
divide ;  and  the  remainder  being  25  half-feet,  is  equal  to  one  half  as  many 
feet,  or  12ft.  6in.  We  adopt  the  same  principle  in  dividing  by  69^;  the 
remainder  being  336  sixths  of  miles,  is  equal  to  one  sixth  as  many  miles,  or 
56  miles.  As  half  a  foot  is  equal  to  6  inches  we  add  them  to  the  3  inches, 
and  have  the  9  inches  in  the  answer. 


3.  In  47  miles  how  many  feet  ? 

4.  In  248160  feet  how  many  miles  ? 

5.  In   78deg.    50m.   7fur.   30rd.   5yd.    2fl.    lOin.   how  many 
inches  ? 

6.  How  many  degrees  in  345056794  inches? 


91.  How  do  vou  reduce  degrees  to  inches  ?  The  reason  of  the  operation? 
How  do  vnu  reduce  inches  to  degrees  1  The  reason  for  the  operation  7  How 
do  you  niidti|)ly  by  ^  ?  How  do  you  divide  by  16^  and  find  the  true  remain- 
der?    How  do  you  obtain  the  true  answer  in  examples  of  this  kind  ? 


REDUCTION. 


93 


SURVEYORS'   MEASURE. 

92.   Surveyors'  Measure  is  used  by  surveyors  in  measuring  land, 

roads,  &c. 

TABLE. 


25 

100 

10 

8 

-  Indies  (in.) 
Links 

Links,  4  Poles,  or  66  Feet, 
Chains 
Furlongs,  or  80  Chains, 

make 

u 
<( 

Link, 

Pole, 

Chain, 

Furlong 

Mile, 

1 

1. 

cha. 

fur. 

m. 

P- 

cha.                        1 

L 

1 
25 

^ 

in. 
198 

fur.                   1=4 

= 

100 

— 

792 

m. 

1        =        10        =          40 

=z 

1000 

— 

7920 

1        = 

=       8       =       80       =       320 



8000 

— 

633G0 

Note.  —  Gunter's  chain,  in  length  4  poles,  or  66  feet,  and  divided  into 
100  links,  is  that  mostly  used  in  ordinary  land  surveys  ;  but  in  locating  roads, 
and  like  public  works,  an  engineer's  chain  is  usually  100  feet  in  length,  con- 
taining 120  links,  each  10  inches  long. 


Mental  Exercises. 

1.  In  2  poles  Hoav  many  links  ?     In  4  poles  ?     In  7  poles  ? 

2.  In  5  chains  how  many  links  ?     In  8cha.  ?     In  lOcha.  ? 

3.  How  many  poles  in  50  links  ?     In  751.  ?     In  1251.  ? 

Exercises  for  the  Slate. 

1.  How  many  links  in  7m.         2.  In  61630  links  how  many 
5fur.  6cha.  301.  ?  miles  ? 


OPERATION. 


OPERATION. 


7  m.  ofur.  6cha.  301. 
8 


6  1  furlongs. 
10 


6  16  chains. 
100 


100)616301. 

10)616  cha.  30i. 
8)61  fur.  6cha. 
7  m.  5  fur. 
Ans.  7m.  5fur.  6cha.  301. 


616  3  0  links,  Ans. 


3.  How  many  miles  in  4386  chains  ? 


^2.  For  what  is  surveyors'  measure  used  1  Recite  the  table.  How  do  you 
reduce  miles  to  links  1  The  rea=!on  for  the  operation  1  How  do  you  reduce 
inches  to  chains  1     To  miles  1     The  reason  of  the  operation  ? 


94 


REDUCTION. 


4.  In  54m.  66cha.  how  many  chains? 

5.  In  75m.  49cha.  how  many  poles  ?    ' 

6.  How  many  miles  in  24196  poles  ? 

7.  How  many  links  in  7m.  4fur.  30rd.  ? 

8.  How  many  miles  in  60750  links  ? 

SURFACE  OR  SQUARE  JIEASURE. 
93.  Square  Measure  is  used  in  measuring  surfaces  of  aU  kinds. 


TABLE. 


144    Square  inches 

make 

1  Square  foot, 

ft. 

9    Square  feet 

(( 

1  Square  yard, 

yd. 

30^  Square  yards,  or  ) 
2721  Square  feet,         ) 

11 

1  Square  rod  or  pole. 

P- 

40    Square  rods  or  poles 

(( 

1  Rood, 

R. 

4    Roods,  or  160  poles, 

(( 

1  Acre, 

A. 

640    Acres 

« 

1  Square  mile, 

ft. 
yd.                     1  = 

S.Jkl 

in. 
144 

P- 

1=               9  = 

1296 

B. 

1 

= 

30^=          2721  = 

39204 

A.               1  = 

40 

— 

1210=        10890  = 

1568160 

S.  M.         1  =        4  = 

160 

= 

4840=        43560  = 

6272640 

1  =  640  =  2560  =  102400 

=  3097600  =  27878400  =  4014489G00 

Note.  —  A  square  is  a  figure  having  four  equal  sides  and  four  equal  angles. 

When  the  four  lines  are  each  1  foot  in  length,  the  space  enclosed  is  1  square 
foot;  when  1  yard  in  length,  1  square  yard ;  when  1  rod  in  length,  1  square 
rod;  and  so  for  any  other  dimension. 


3ft.  =  Ivd. 


In  this  diagram  the  larfje  square  represents 
a  square  yard,  and  each  of  the  smaller  squares 
within  it  represents  one  square  foot.  Now, 
since  there  are  three  rows  of  small  squares, 
and  three  square  feet  in  each  row,  there  will 
be  3  sq.  ft.  X  3  =  9  sq.  ft.  in  the  larn;e 
square.  But  the  large  square  is  3  ft.  m 
length  and  3  ft.  in  breadth ;  hence, 


To  find  the  contents  of  a  square,  multiply  the  numbers  denoting  its 
length  and  breadth  together. 


Square 
foot. 

93.  For  what  is  square  measure  used  ?  Repeat  the  table.  "Wliat  te  a 
square?  A  square  foot?  How  may  the  contents  of  a  square  be  found? 
Explain  by  the  diagram  the  reason  of  the  operation. 


REDUCTION.  95 

Mental  Exercises. 

1.  In  2  square  feet  how  many  square  inches  ? 

2.  In  3  square  yards  how  many  square  feet  ?     In  10  sq.  yd.  ? 

3.  In  5  roods  how  many  poles  ?     In  20  roods  ?     In  30  roods  ? 

4.  In  7  acres  how  many  roods  ?      In  24  acres  ?     In  40  acres  ? 

Exercises  for  the  Slate. 
1.  How  many  square  inches  in  12A.  3R.  24p.  144ft.  72in.  ? 

OPERATION. 

1  2  A.  3R.  24p.  144ft.  72in. 
4 


5  1  roods. 
40 


2  0  6  4  poles.  Note.  —  To  multiply  by  ^,  we 

2  7  2  1  ^^^^  k  of  the  multiplicand. 


4132 
14452 
4129 

516 

6  6  2  0  6  8  feet. 
144 


2248274 
2248279 
562068 


Ans.  80937864  inches. 

2.  In  80937864  square  inches  how  many  acres? 

OPERATION. 

144)80937864  inches. 

2  7  2^)562068  ft.  72m. 
4  4 


1  0  8' 9  )2248272  fourths  of  a  foot. 

40)2064  poles.     576  -=-  4  =  144fl. 
4)  5  1  R.  24p. 
Ans.  1  2  A.  3R.  24p.  144ft.  72in. 

Note.  —  To  divide  by  the  272^,  we  first  reduce  the  divisojj  and  dividend 
to  fourths,  and  then  divide.  The  remainder  obtained,  being  fourths,  is  reduced 
to  whole  numbers  by  dividing  by  4. 

*  93.  How  do  you  reduce  acres  to  square  inches  ?  The  reason  for  the  opera- 
tion? How  do  you  reduce  square  inches  to  acres'?  The  reason  for  the 
operation  ?     How  do  you  multiply  by  4 1 


96  REDUCTION. 

3.  In  49  A.  3R.  16p.  how  many  square  feet  ? 

4.  In  2171466  square  feet  how  many  acres? 

5.  What  is  the  value  of  3  65 A.  3R.  17p.  at  $  1.75  per  square 
rod  or  pole  ?  Ans.  $  102,439.75. 

6.  Sold  a  valuable  piece  of  land,  containing  3 A.  IR.  30p.,  at 
$  1.25  per  square  foot ;  what  was  received  for  the  land  ? 

Ans.  $187,171,875. 

7.  In  a  tract  of  land  12  miles  square,  how  many  square  mUes? 
How  many  acres?  Ans.  92160  acres. 

8.  In  18 A.  OR.  16p.  how  many  square  feet  ? 

Ans.  788436  square  feet. 

9.  Purchased  48A.  3_R.  14p.  of  land  for  $2.25  per  square 
rod,  and  sold  the  same  for  $3.15  per  square  rod  ;  what  did  I 
gain  by  my  bargain  ?  Ans.  $  7032.60. 

CUBIC   OR  SOLID  IIEASURE. 

94.  Cubic  or  Solid  Measure  is  used  in  measuring  such  bodies 
or  things  as  have  length,  breadth,  and  thickness  ;  as  timber, 
stone,  &c. 

TABLE. 


1728  Cubic  inches  (cu.  in.) 

make 

1  Cubic  foot, 

cu.  ft. 

27        "     feet 

u 

1       "     yard. 

cu.  yd. 

40        "     feet 

(( 

1  Ton, 

T. 

16        "     feet 

u 

1  Cord  foot, 

eft. 

8  Cord  feet,  or  > 
128  Cubic  feet,      | 

(( 

1  Cord  of  wood, 
ft. 

C. 

in. 

yd. 

1          = 

1728 

T. 

1 

r=                 27              =. 

46656 

a                 1          = 

H 

4 

=            40          = 

69120 

1       ==       Si        = 

A 

^ 

=          128          = 

221184 

Note  1.  —  A  pile  of  wood  8ft  in  length,  4ft.  in  breadth,  and  4ft.  in  height, 
contains  a  cord. 

One  ton  of  timber,  as  usually  surveyed,  contains  ^O^i^  cubic  feet. 

A  perch  of  masonry  is  IG^ft.  long,  1ft.  high,  and  l^ft.  thick,  or  24|  cubic 
feet. 


93.  How  do  you  divide  by  272  J  ?  Of  what  denomination  is  the  remainder  ? 
How  is  the  true  rcmaindor  found  ? — 94.  For  wliat  is  cnhic  measure  u-^cd  ■? 
Recite  the  table.  What  are  the  dimensions  of  a  pile  of  wood  containing 
1  cord  ?  How  many  solid  feet  does  a  ton  of  limber  contain,  as  usually 
surveyed  7 


UEDUCTION. 


97 


KoTE  2.  —  A  cube  is  a  solid  bounded  by  six  square  and  equal  sides. 

If  tbc  oul)e  is  1  foot  loiiu:,  1  foot  wiilC;  and  1  foot  bigb,  it  is  called  iiaihic 
or  aol III  foul.  If  t!ie  fnl)C  is  3  feet  ioii^',  3  f^et  wide,  and  3  f^et  tliielv,  it  is 
called  a  cubic  or  solid  i/ard. 

Since  each  side  of  a  yard,  as  repre- 
sented in  the  diagram  (Art.  i)'3),  con- 
tains 9  scj.  ft.  of  surface,  it  is  plain,  tliat 
if  a  block  be  cut  off  from  one  side,  one 
foot  thick,  it  can  be  divided  into  9  t^olid 
blocks,  with  sides  1  .foot  in  lenjrth, 
breadtli,  and  thickness,  and  therefore 
•will  contain  9  solid  I'eet;  and  since  the 
■whole  block  or  cube  is  lln-ee  feet  thick, 
it  must  contain  9  solid  ieet  X  3  =  27 
solid  feet;  or  3  solid  feet  X  3  X  3 
=  27  solid  feet.     Hence, 

■    7b  Ji/id  the  contents  of  a  cubic  bod//,  multiphj  together  the  numbers 
denoting  the  length,  breadth,  and  thichicss. 

Mental  Exercises. 

1.  In  2  cubic  feet  how  many  cubic  inches?     In  4  cu.  ft.? 

2.  In  3  cubic  yards  how  many  cubic  feet?     In  10  cu.  yd.? 

3.  How  many  cords  of  Avood  in  G4  cord  feet?     In  9G  c.  ft.  ? 

4.  How  many  tons  in  80  cu.  ft.  of  timber  ?     In  IGO  cu.  ft.  ? 

Exercises  for  the  Slate. 

2.  In  2265408  cubic  inches 


3  ft.  =  1  J  d. 


1.  In  48  cu.  yd.  and  15  cu 
ft.  how  many  cubic  inches  ? 

OPETtATION. 

4  8  yd.  15ft. 
27 


341 
97 


how  many  cubic  yards  ? 

opehatiox. 
17  2  8  )  2  2  6  5  4  0  8  cu.  in. 

2  7)1311  cu.  ft. 
Ans.  4  8  yd.  loft. 


13  11  feet 
1728 


10488 
2G22 
9177 
13  11 


Ans.  2  2  6  5  4  0  8  inches. 


94.  Wlmt  is  a  cube  ?   ITow  do  yon  find  tlic  contents  of  a  cube  1  Tbc  reason 
for  tbp  oneration.     Descrilie  a  cubic  foot.     How  do  vou  reduce  a  ton   to 
cubic  inches'     Tbe  reason  for  the  operation.     How  do  you  reduce  cubic 
inches  to  cubic  yards  ?     The  reason  for  the  operation. 
9 


98  KEDUCTION. 

3.  In  45  cords  of  wood  how  many  cubic  inches  ? 

4.  In  9953280  cubic  inches  how  many  cords  of  wood  ? 

5.  How  many  cubic  feet  in  a  pile  of  wood  15ft.  long,  4ft.  wide, 
and  GJ^ft.  high  ?     How  many  cords  ?  Ans.  3C.  6  cu.  ft. 

6.  How  many  cubic  inches  in  a  block  of  marble  4ft.  long  3^ft. 
wide,  and  2ft.  thick  ?  Ans.  44928. 

7.  In  a  room  14ft.  long,  12ft.  wide,  and  8ft.  high,  how  many 
cubic  feeK^        .  Ans.  1344. 

8.  What  will  9080  cubic  feet  of  ship-timber  cost,  at  $  11.50 
per  ton?  Ans.   $2610.50. 

"WINE   OR  LIQUID  MEASURE. 

95.    Wine  or  liquid  Measure  is  used  in  measuring  all  kinds  of 
liquids,  except,  in  some  places,  beei-,  ale,  porter,  and  milk. 

TABLE. 


4  Gills  (gl.) 

make 

Pint, 

pt. 

2  Pints 

(( 

Quart 

. 

qt. 

4  Quarts 

(( 

Gallon, 

gal. 

63  Gallons 

(( 

Hogshead, 

hhd. 

2  Hogsheads 

.(( 

Pipe, 

or  Butt 

'1 

pi- 

2  Pipes 

(( 

qt. 

Tun, 

pt. 
1 

tun. 

gi- 
4 

gal. 

1 

= 

2 



8 

hbd. 

1 

= 

4 

= 

8 

^= 

32 

p5. 

1      = 

63 

= 

252 

= 

504 



2016 

tun.             1       = 

2      = 

126 

= 

504 

= 

1008 

1= 

4032 

1      =      2      = 

4      = 

252 

= 

1008 

=: 

2016 

= 

8064 

Note  1.  —  In  some  States  31 J  gallons,  and  in  otliers  from  28  to  32  gal- 
lons, make  1  barrel.     42  gallons  make  1  tierce,  and  2  tierces  1  puncheon. 

Note  2. —  The  term  hogshead  is  often  applied  to  any  large  cask  that  may 
contain  from  50  to  120  gallons,  or  more. 

Note  3.  —  The  Standard  Unit  of  Liquid  Measure  adopted  by  the  povern- 
ment  of  the  United  States  is  the  \Viiicli<ster  Wine  Gallon,  which  contains  •.:31 
cubic  inches.  It  has  the  name  Winchester,  from  its  standard  havintr  been 
foi-merly  kept  at  Winchester,  En<rhind.  Tlie  Imperial  Gallon,  now  adopted 
in  Great  Britain,  contains  277^V(nr  <^"'''<^  inches  ;  so  that  6  Winchester 
gallons  make  about  5  Impeiial  gallons. 

95.  For  what  is  wine  or  liquid  measure  used  ?  Eepeat  the  table.  ITow 
many  frallons  make  n  barrel  ?  A  tierce?  A  puncheon  ?  How  i<  the  term 
Logshcad  often  applied  ?     What  is  the  standard  unit  of  liquid  measure  1 


KEDUCTION. 


99 


Mental  Exercises. 

1.  In  3  pints  how  many  gills?     In  5  pints?     In  9  pints? 

2.  In  4  quarts  how  many  pints  ?     In  6  quarts  ?     In  8  quarts  ? 

3.  In  5  gallons  how  many  quarts  ?     In  7  gallons  ? 

4.  How  many  gallons  in  12  quarts?     In  18  quarts? 


1.  In   47  tuns   of   wine  how 
many  gills  ? 

OPERATION. 

4  7  tuns. 
4 


Exercises  for  the  Slate. 

2.  In   379008 -gills   how 
many  tuns? 

OPERATION. 

4)379008  gi. 


18  8  hogsheads. 
63 


564 
1128 

118  4  4  gallons. 
4 


2)94752  pt. 
4)47376  qt. 
6  3  )  1  1  844  gal. 
4)188  hhd. 
Ans.  4  7  tuns. 


4  7  3  7  6  quarts. 
2 


9  4  7  5  2  pints. 
4 


Ans.  3  7  9  0  0  8  gills. 

3.  Reduce  197  tuns  3hhd.  60gal.  3qu.  Ipt.  to  gills. 

4.  In  1596604  gills  how  many  tuns  ? 

5.  What  will  7  hogsheads  of  wine  cost,  at  5  cents  a  pint  ? 

6.  What  cost  18  tuns  Ihhd.  47gal.  of  oil,  at  $1.25  per  gal- 
lon ?  Ans.  $  5307.50. 

BEER  MEASURE. 

96.   Beer  Measure  is  used  in  measuring  beer,  ale,  porter,  and 
tailk. 

TABLE. 

qt. 

gal. 

hhd. 

pt. 

2 

8 

432 


2  Tints  (pt.) 
4  Quarts 
54  Gallons 

make 

1  Quart, 
1  Gallon, 
1  Hogshead, 

qt. 

hhd. 

gal. 
1 

1 

4 

1               = 

54 

= 

•216 

96.  Repeat  the  tabic  of  beer  measure. 


100  REDUCTION. 

Note  1.  —  The  gallon  of  beer  measure  contains  282  cubie  inches;  and 
as  has  been  usually  reckoned,  36  gallons  equal  1  barrel ;  2  hogsheads,  or 
108  gallons,  1  l)Utt ;  2  butts,  or  216  gallons,  1  tun.  1  galbn  beer  measure 
=  Igall.  Ipt.  3/ygi-  wine  measure. 

Note  2.  —  Beer  Measure  is  becoming  obsolete.  Milk  and  malt  liquors,  at 
the  i)resent  time,  are  bought  and  sold,  generally,  by  wine  or  liquid  measure. 

9  Exercises  for  the  Slate. 

1.  How   many   quarts  in  76  2.  In   1G416   quarts  how 

hogsheads  ?  many  hogsheads  ? 

OPERATION.  OPERATION. 

7  6  hhd.  4  )  1  6  4  1  6  qt. 

54 


304 
380 

410  4  gallons. 
4 


5  4)4104  gal. 
Ans.  7  6  hhd. 


Ans.  16  4  16  quarts. 

3.  In  4  tuns  Ihhd.  17gal.  Ipt.  how  many  pints  ? 

4.  How  many  tuns  in  7481  pints? 

5.  What  cost  7hhd.  18gal.  of  beer  at  4  cents  a  quart? 

Ans.  $  63.36. 

6.  At  15  cents  per  gallon,  what  will  IShhds.  of  ale  cost? 

Ans.  $  145.80. 
DRY  MEASURE. 

97.    Dry  Measure  is  used  in  measuring  grain,  fruit,  salt,  coal,  &c. 

TABLE. 


2  Pints  (pt.) 

make 

1   Quart, 

qt. 

8  Quarts 

(( 

1   Peck, 

pk. 

4  I'ccks 

i( 

1  Bushel, 

bu. 

8  Bushels 

(( 

1  Quarter, 

qr. 

Cll. 

SG  Bushcb 

t( 

1  Chaldron, 

qt. 

pt. 

pk. 

1 

— 

2 

bo. 

1 

=               8 

= 

.^6 

ch. 

1 

=             4 

=              32 

= 

64 

1           == 

86 

=          144 

=          1152 

== 

2304 

96.  How  many  cubic  inches  does  the  boor  gallon  contain  ?  How  do  you 
reduce  hogsheads  to  quarts?  Quarts  to  Iiogshcado  7  —  97.  For  what  is  dry 
measure  used  ?     Repeat  the  table. 


REDUCTION.  101 

Note  1.  —  Tlie  Standard  Unit  of  Dri/  Measure  adopted  by  tlie  United 
States  government  is  the  Winchester  bushel,  which  is  in  form  a  cylinder,  18^ 
inches  in  diameter,  and  8  inches  deep,  containing,  2150^^5^0-  cubic  inches. 
The  Slandiird  Imperial  Bushel  of  Great  Britain  contains  •22\8.^^^g  cut)ic 
inches,  so  that  32  Imperial  bushels  equal  about  33  Winchester  bushels.  The 
gallon  in  Dry  Measure  contains  268^  cubic  inches. 

Note.  2. —  Igal.  Dry  Measure  =  268fcu.  in.  =  Igal.  Ipt.  l^fgi.  "Wine 
Measure  =  3qt.  l^lpt.  Beer  Measure  ;  Igal.  W.  M.  =  23 leu.  in.  =  3qt. 
Ipt.  D.  M.  =  3qt.  flpt.  B.  M. ;  Igal.  B.  M.  =  282cu.  in.  =  Igal.  Ipt. 
sAgi.  W*  M.  =  Igal.  ^Jpt.  D.  M.  ;  Iqt.  D.  M.  =  67|cu.  in.  =  Iqt. 
iHyi-  W.  M.  ;  Iqt.  W.  M.  =  57|cu.  in.  =  iffpt.  D.  M. ;  Ipt.  D  M. 
=  33|cu.  iu.  =  Ipt.  ]|gi.  W.  M. ;  Ipt.  W.  M.  =  isjcu.  in.  =  ||pt.  D.  M. 

Mental  Exercises. 

1.  In  2  quarts  how  many  pints?     In  5  quarts?     In  7  quarts? 

2.  In  3  pecks  how  many  quarts  ?     In  6  pecks  ?     In  9  pecks  ? 

3.  In  5  bushels  how  many  pecks?     In  10  bushels? 

4.  'How  many  pecks  in  16  quarts?     In  25  quarts? 

Exercises  for  the  Slate. 

1.  How  many  quarts  in  2.  In  5G731   quarts  how 

49ch.  8bu.  3pk.  and  3qt.  ?  many  chaldrons  ? 

OPERATION.  OPERATION. 

4  9  eh.  8bu.  3pk.  3qt.  8  )5  673  1  qt. 

^  ^  4  )  7  0  9  1  pk.  3qt. 

3  6)1772  bu.  2pk. 
4  9  eh.  8bu. 

Ans.  49ch.  8bu.  3pk.  3qt. 


Ans.  5  6  7  3  1  quarts. 

3.  Reduce  97ch.  30bu.  2pk.  to  quarts. 

4.  In  112720  quarts  how  many  chaldrons? 

5.  How  many  pints  in  35bu.  Ipt.  ? 

6.  Reduce  2241  pints  to  bushels. 

7.  Reduce  18qr.  3pk.  5qt.  to  quarts. 

8.  How  many  quarters  in  4637  quarts? 

9.  In  19bu.  3|)k.  7qt.  Ipt.  how  many  pints? 
10.  In  1279  pints  how  many  bu-hels  ? 

97.  Wh.1t  is  the  standard  unit  of  Dry  Measure? 
9* 


302 
147 

17  7  2  bushels. 
4 

7  0  9  1  pecks. 
8 

102 


REDUCTION. 


MEASURE   OF   TIME. 

98.    Measure  of  Time  is  applied  to  the  various  divisions  and 
sub-divisions  into  which  time  is  divided. 


TABLE. 


GO  Seconds  (sec.) 

make 

Minute, 

m. 

60  Minutes 

n 

Hour, 

•    t. 

24  Hours 

(( 

Day, 

da. 

7  Days 

(( 

Week, 

■w. 

365^  Days,  or  52  weeks    } 
H  days,                ; 

(( 

Julian  Year, 

y- 

12  Calendar  Months  (mo.) 

(1 

Year, 

m. 

y- 

sec. 

,      h. 

1      = 

CO 

d. 

1 

= 

GO     = 

3600 

yr.                        1        = 

24 

— 

1440     = 

€6400 

y.                  1          =           7       = 

1G8 

= 

10080     = 

C04800 

1      =      52^^      =      365^     = 

87G6 

= 

525960     = 

31557600 

Note  1  ■  —  The  true  Solar  or  Tropical  Year  is  the  time  measured  from  the 
sun's  leaving  either  equinox  or  solstice  to  its  return  to  the  same  again,  and 
is  365d.  5h.  48m.  49sec.  nearly. 

The  Julian  Year,  so  called  from  the  calendar  instituted  by  Julius  Ciesar, 
contains  36.');i  days,  as  a  medium  ;  three  years  in  succession  containing  365 
days,  and  the  fourth  year  366  days  ;  which,  as  compared  with  the  true  solar 
year,  produces  a  yearly  error  of  11m.  10  j%  sec,  or  of  1  whole  day  in  about 
120  years. 

The  Gregorian  Year,  or  that  instituted  bv  Pope  Gregory  Xin.,  in  the  year 
1582,  and  which  is  now  the  Cinil  or  Le<jal  Year  in  use  among  the  ditfercnt 
nations  of  the  earth,  contains  365  days  for  three  years  in  succession,  and  366 
days  for  the  fourtli,  exceptiiu/  centennial  i/ears  ivliose  number  cannot  be  exactly 
divided  bjj  400.  The  Gregorian  year  gives  an  error  of  only  1  day  in  3866 
years. 

A  Common  Year  is  one  of  365  days,  and  a  Leap  or  Bissertile  Year  is  one 
of  366  days.  Any  year  is  Leap  Year  whose  number  can  be  divided  by  4 
wiiliout  a  remaincler,  except  years  whose  number  can  be  divided  without  a 
remainder  by  100,  but  not  by  400. 

A  Sidereal  Year  is  the  time  in  which  the  earth  revolves  round  the  sun,  and 
is  365d.  6h.  9m.  g/^sec. 

Note  2.  —  The  12  calendar  months,  composing  the  civil  year,  arc  Jan- 
uary,  Fei)ru;iry,  March,  April,  May,  June,  July,  August,  September,  October, 
November,  December,  anil  the  number  of  days  in  each  may  be  readily  re- 
meml)ercd  by  the  following  lines  :  — 


98.  To  what  is  the  measure  of  time  applied  ?  Repeat  the  table.  How  is 
tlie  true  solar  year  measured  '?  How  long  is  it?  Why  is  the  .Julian  year  so 
called?  Who  instituted  the  Gregorian  year?  What  is  a  Common  year ? 
A  Sidereal  year  ?     Name  the  months  iu  their  order. 


EEDUCTION. 


103 


"  Thirty  days  hath  September, 
AprilJ  Juiie,  and  November; 
And  all  the  rest  luive  thirty-one, 
Save  February,  which  alone 
Hath  twenty-eight;  and  this,  in  fine, 
One  year  in  four  hath  twenty-nine." 

TABLE 

Sho'WIng  the  Number  of  Days  froji  any  Day  of  one  Month  to  the 
SAJiE  Day  of  any  othek  Month  in  the  same  Year. 


To  THE  SAME  Day  of 

From  axt 
Dat  of 

Jan. 

Feb. 

Mar. 

Apr. 

May. 'June. 

July. 

Aug. 

Sept. 

Oct. 
273 

Nov. 
304 

Dec. 
334 

January 

365 

31 

59 

90 

120 

151 

181 

212 

243 

February 

3.34 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

March 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

April 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

May 

245 

276 

304 

335 

365 

31 

61 

92 

123 

1.53 

184 

214 

June 

214 

245 

273 

304 

334 

365 

30 

61 

92 

122 

153 

183 

July 

184 

215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

1.53 

August 

153 

184 

212 

243 

273 

304 

334 

365 

31 

61 

92 

122 

September 

122 

153 

181 

212 

242 

273 

303 

334 

365 

30 

61 

91 

October 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

November 

61 

92 

120 

151 

181 

212 

242 

273 

304 

334 

365 

30 

December 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 

For  example,  to  find  the  number  of  days  from  April  4th  to  November  4th, 
vra  look  for  April  in  the  left  vertical  column,  and  November  at  the  top,  and, 
where  tho  lines  intersect,  is  214,  the  number  souq:ht.  A5:ain,  to  find  the 
number  of  days  from  June  10th  to  September  16th,  we  find  the  difference 
between  June  10th  and  September  10th  to  be  92  days,  and  add  6  days  for  the 
excess  of  the  16th  over  the  10th  of  September,  so  we  have  98  days  as  the 
exact  dift'erence. 

If  the  end  of  Febraary  be  included  between  the  points  of  a  time,  a  day 
must  be  adiled  in  leap  year. 

When  the  time  exceeds  one  year,  there  must  be  added  365  days  for  each 
year. 

Mental  Exercises. 


1. 
2. 

o 
O. 

4. 
5. 
6. 


In  3  minutes  how  many  seconds  ?     In  5  minutes  ? 

In  2  hours  how  many  minutes  ?     In  4  hours  ? 

In  4  weeks  how  many  days  ?     In  6  weeks  ?     In  9  weeks  ? 

In  2  days  how  many  hours  ?     In  3  days?     In  7  days? 

How  many  weeks  in  21  days?     In  30  days?     In  50  days? 

How  many  calendar  months  in  2  years  ?     In  8  years  ?     In 


10  years  ?     In  12  years  ?     In  20  years"? 


98.  How  many  days  ha"  each  month  ■?  How  do  yon  find  by  the  table  the 
number  of  days  from  April  4th  to  November  4th  '?  "  Wli^n  the  time  sought 
for  is  more  than  one  year,  how  many  days  muit  be  added  1 


1405 
730 

8  7  6  5  hours. 
60 

5  2  5  9  4  8  minutes. 
GO 

104  REDUCTION. 

Exercises  for  the  Slate. 

1.  How  many  seconds  in  3G5cla.        2.  In  3155G929   seconds 
5h.  48m.  49sec.,  or  one  solai'  yeai*  ?    how  many  days  ? 

OPERATION.  OPERATION. 

3  G  5  da.  5h.  48m.  49sec.  6  0)  3  1  5  5  G  9  2  9 

ii  6  0)  5  2  o  9  4  8m.  49sec. 

2  4)  8  7  6  5  h.  48m. 
3  6  5  da.  5h. 
Ans.  365da.  5h.  48m.  49sec. 


31556929  seconds,  Ans. 

3.  Reduce  29Gda.  18h.  32m.  to  minutes. 

4.  In  427352  minutes  how  many  days? 

5.  How  many  seconds  in  30  solar  years  262da.  17h.  28m. 
42sec.  ? 

6.  In  969407592  seconds  how  many  solar  years? 

7.  How  many  weeks  in  G84592  minutes  ? 

8.  In  G7w.  6d.  9h.  52m.  how  many  minutes  ? 

9.  How  many  days  from  June  5th  to  Dec.  11th  ? 

10.  How  many  days  from  March  17th,  185G,  to  May  16th, 
1857  ?  Ans.  425  days. 

11.  How  many  days  from  December  18th,  1856,  to  January 
30th,  1857  ? 

12.  How  many  days  from  August  30th,  1857,  to  June    1st, 
1858? 

13.  How   many    days    from   July   4th,    1859,    to   July   4th, 
18G0? 

14.  How  many  days  from  April  25tb,  1855,  to  August  20lh, 
1858?  Ans.  1213  days. 

K'oTK.  —  The  last  six  examples  arc  to  be  performed  by  aid  of  tlie  table  on 
page  103. 


98.  How  do  you  reduce  years  to  seconds?  The  reason  for  the  opemtinn. 
IIo\r  do  you  rcduco  seconds  to  days  ?  To  years  ?  The  reason  for  the 
operation. 


REDUCTION. 


105 


CIRCULAR  MEASURE. 

99.  Circular  Measure  is  applied  to  the  measurement  of  circles 
and  angles,  and  is  used  in  reckoning  latitude  and  longitude,  and 
the  revolutions  of  the  planets  round  the  sun. 

TABLE. 


60  Seconds  (") 

60  Minutes 

30  Degrees 

1 2  Signs,  or  360  Degrees, 

make 

u 
« 

1  ^linute,                                      '. 
1  Degree,                                     °." 
1  Sign,                                             S. 
The  Circle  of  the  Zodiac,          C. 

S. 
C.                       1         = 
1         =         12         = 

o 

1 

30 

360 

1 

=               60 
=           1800 
=         21600 

==                   60 
=               3600 
=           108000 
=         1296000 

£« 


©"fcr 


081 


Note  1.  —  A  Circle  is  a  plane  figure 
bounded  by  a  curve  Une,  every  part  of 
which  is  equally  distant  from  a  point 
called  its  center. 

The  Circumference  of  a  circle  is  the  line 
■which  bounds  it,  as  shown  by  the  diagram. 
An  Ai-c  of  a  circle  is  any  part  of  its  cir- 
cumference ;  as  AB. 

A  Radius  of  a  circle  is  a  straight  line 
drawn  from  its  center  to  its  circumference  ; 
as  CA,  CB,  or  CD. 

Every  circumference  is  supposed  to  be 
divided"  into   360  equal   parts,  called   de- 
grees. 
A  Quadrant  is  one  fourth  of  a  circumference,  or  an  arc  of  90° ;  as  AB. 
An  Ani/le,  as  ACB,  is  the  inclination  or  opening  of  two  lines  which  meet 
at  a  point,  as  C      The  point  is  the  vertex  of  the  angle.     If  a  circumference 
be  drawn  around  the  vertex  of  an  angle  as  a  center,  the  two  sides  of  tlie 
angle,  as  radii  of  the  circle,  will  include  an  arc,  which  is  the  measure  of  the 
angle  ;  as  the  arc  AD  =  120°  is  the  measure  of  the  angle  ACD,  and  AB 
=  90°,  the  measure  of  the  augle  ACB  ;  hence  the  one  is  an  angle  of  120°, 
and  the  other,  of  90°. 

Note  2.  —  As  the  earth  turns  on  its  axis  from  west  to  east  every  24  hours, 
the  sun  appears  to  pass  from  east  to  west  -j^  of  .360°  of  longitude  every  hour, 
or  over  1.5°  of  longitude  in  1  hour's  time,  or  1°  in  4  minutes  of  time,  and  1' 
in  4  seconds  of  time  ;  so  that  when  it  is  noon  at  any  place,  it  is  1  hour  earlier 
for  every  15°  of  longitude  westward,  and  1  hour  later  for  every  15°  of  longi- 
tude eastward.  Thus,  -Boston  being  71°  4'  west  of  Greenwich,  and  San 
Francisco  51°  17'  we^t  of  Boston,  when  it  is  noon  at  Boston,  it  is  4h.  41m. 
16sec.  past  noon  at  Greenwich,  and  wanting  3h.  25m.  8sec.  of  noon  at  Saij 
Francisco. 


99.  To  what  is  circular  measure  applied  ?     Recite  the  table, 
circle  ?     An  angle  1 


What  is  a 


106 


E£D¥CTION. 


Exercises  for  the  Slate. 


1.  IIow    many    minutes    ia 
lis.  18°  57'? 

OPEnATION. 

1  1  S.  18°  57' 
30 


2.  In    20937    minutes 
bow   many   signs  ? 

OPERATION. 

G  0  )  2  0  9  3  7' 


3  4  8  degrees. 
GO 


3  0  )  3  4  8°  5  7' 


Ans.  2  0  9  3  7  minutes. 

3.  In  27S.  19°  51'  28"  how  many  seconds? 

4.  How  many  signs  in  2987488  seconds.'* 


1  1  S.  1  8° 
Ans.  lis.  18°  57'. 


MISCELLANEOUS  TABLE. 

100.    This   table   embraces   a  variety   of  denominations   fre- 
quently used  in  business. 


1 2  units 
12  dozen 
1 2  gross 
2U  units 
14  pounds 
60  pounds 
GO  pounds 
60  pounds 
60  pounds 
52  pounds 
70  pounds 
.56  pounds 
6G  pounds 
56  pounds 
45  pounds 
20  pounds 
48  pounds 
62  pounds 
4fi  pounds 
82  pounds 
80  pounds 


9fl.  TIow  do  you  reduce  siorns  to  seconds  ?  Oivc  the  reason  of  the  operation. 
How  do  you  reduce  .iicconds  to  dc^rrces  ?  To  si;;ns  ?  Give  the  rca'iim  for 
the  operation.  How  many  degrees  in  a  circle  ?  —  100.  What  is  embraced  in 
the  miscellaneous  tah'.e  t 


make 

1  dozen. 

u 

1  gross. 

(( 

1  great  gross. 

l( 

1  score. 

of  Iron  or  Lead 

it 

1  stone. 

of  Wheat 

(1 

1  bushel. 

of  Clover-seed 

(t 

1  bushel. 

of  Beans 

^l 

1  bushel. 

of  Potatoes 

(t 

1  bushel. 

of  Onions 

II 

1  bushel. 

of  Corn  on  the  Cob 

u 

1  bushel. 

of  Shelled  Corn 

11 

1  bushel. 

of  Rye 

II 

1  bushel. 

of  Flax-seed 

11 

1  bushel. 

of  Timothy-seed 

II 

1  bushel. 

of  l^ran 

it 

1  bushel. 

of  Barley 

(( 

1  bushel. 

of  Buckwheat 

II 

1  bushel  in  Ky. 

of  Buckwheat 

11 

1  bushel  in  ]\Iass.  and  Pa. 

of  Oats 

It 

1  l)ushcl  in  INIass..  111.,  O.,  etc. 

of  Oats 

11 

1  bushel  in  Me.,  N.  H.,  Pa.,  et 

I 


■\ 


REDUCTION.  107 

196  pounds  of  Flour  make  1  barrel. 

200  pounds  of  Beef  "  1  barrel. 

200  pounds  of  Pork  «  1  barrel. 

100  pounds  of  Fish  "  1  quintal. 

200  pounds  of  Shad  or  Salmon   "  1  barrel  in  N.  Y.,  Ct. 

220  pounds  of  Fish  "  1  barrel  in  Md. 

30  gallons  of  Fish  "  1  barrel  in  Mass. 

5  bushels  of  Corn  "  1  barrel  in  Md.,  Tenn.,  etc. 

24  sheets  of  Paper  make  1  Quire. 

20  (quires  "  1  Keam. 

2  reams  "  1  Bundle. 

5  bundles  "  1  Bale. 

Note.  —  A  sheet  folded  in  2  leaves  forms  a  folio  ;  in  4  leaves,  a  quarto  ; 
in  8  leaves,  an  octavo;  in  12  leaves,  a  12mo  ;  in  18  leaves,  an  ISmo;  and 
in  24  leaves,  a  24mo. 

mSCELLANEOUS   EXERCISES. 

1.  In  $  315.18how  many  mills  ? 

2.  How  many  dollars  in  345180  mills  ? 

3.  In  46£  IBs.  5d.  how  many  farthings  ? 

4.  How  many  pounds  in  45044  farthings  ? 

5.  Reduce  611b.  Ooz.  17pwt.  17gr.  troy  to  grains. 

6.  In  351785  grains  troy  how  many  pounds? 

7.  How  many  scruples  in- 271b  3§  13  19  ? 

8.  In  7852  scruples  how  many  pounds  ? 

9.  In  83T.  llcwt.  3qr.  181b.  how  many  ounces? 

10.  How  many  tons  in  2675088  ounces  ? 

11.  How  many  nails  in  97yd.  3qr.  3na.  ? 

12.  In  1567  nails  how  many  yards  ? 

13.  In  57  ells  English  how  many  yards  ? 

14.  How  many  ells  English  in  71yd.  Iqr.  ? 

^15.  How  many  inches  in  15m.  7fur.  18rd.  10ft.  6in.  ? 
?  16.  In  1009530  inches  how  many  miles  ? 

17.  In  95,000,000  of  miles  how  many  inches  ? 

18.  How  many  miles  in  6,019,200,000,000  inches? 

19.  In  48deg.  18m.  7fur.  18rd.  how  many  feet? 

20.  Ill  17629557  feet  how  many  degrees  ? 

21.  How  many  square  feet  in  7 A.  3R.  16p.  218ft.? 

22.  In  342164  square  feet  how  many  acres  ? 

23.  How  many  square  inches  in  25  square  miles  ? 


100.  What  gives  name  to  the  size  or  form  of  books  1 


108  REDUCTION. 

24.  In  100362240000  square  inches  how  many  square  miles? 

25.  How  many  cubic  inches  in  15  tons  of  timber? 
20.  In  lOoGbUO  cubic  inches  liow  many  tons  ? 

27.  How  many  gills  of  wine  in  5hhd.  17gal.  3qt.  ? 

28.  In  10648  gills  how  many  hogsheads  of  wine  ? 
20.  How  many  quarts  of  beer  in  29hhd.  30gal.  3qt.? 

30.  In  6387  quarts  of  beer  how  many  hogsheads  ? 

31.  How  many  pints  in  15cli.  16bu.  3pk.  of  wheat? 

32.  In  35632  pints  of  wheat  how  many  chaldrons  ? 

33.  How  many  seconds  of  time  in  365  days  6  hours  ? 

34.  In  31557600  seconds  how  many  days  ? 

35.  How  many  hours  in  1842  years  (of  365da.  6h.  each)  ? 

36.  In  16146972  hours  how  many  years  ? 

37.  How  many  seconds  in  8S.  14°  18'  17"  ? 

38.  In  915497"  how  many  signs? 

39.  What  will  be  the  cost  of  13  gross  of  steel  pens,  at  2i  cents 
per  pen  ?  Ans.  $46.80. 

40.  Bought  12  reams  of  paper  at  20  cents  per  quire ;  how 
much  did  it  cost  ?  Ans.  $  48. 

41.  I  wish  to  put  2  hogsheads  of  wine  into  bottles  that  will 
contain  3  quarts  each  ;  how  many  bottles  are  required  ? 

Ans.  168  bottles. 

42.  "When  $1480  are  paid  for  25  acres  of  knd,  what  costs  1 
acre  ?     Wliat  costs  1  rood?     What  cost  37A.  2R.  18p.? 

Ans.  $2226.66. 

43.  John  Web>ter  bought  5cwt.  3qr.  181b.  of  sugar  at  9  cents 
per  lb.,  for  which  he  paid  25  barrels  of  apples  at  $  1.75  per  bar- 
rel; how  much  remains  due  ?  Ans.  $  9.62. 

44.  Bought  a  silver  tankard  weighing  21b.  7oz.  for  $  46.50 ; 
what  did  it  cost  per  oz.  ?     How  much  per  lb.  ?  Ans.  $  18. 

45.  Bought  3T.  Icwt.  181b.  of  leather  at  12  cents  per  lb.,  and 
sold  it  at  9  cents  per  lb.  ;  what  did  I  lose?         Ans.  $  183.54. 

46.  Phineas  Bailey  has  agreed  to  grade  a  certain  railroad  at 
$  5.75  per  rod  ;  wliat  Avill  he  receive  for  grading  tiie  road,  its 
length  being  37m.  7fur.  29rd.  ?  Ans.  $  69856.75. 

47.  If  it  cost  $17.29  a  rod  to  grade  a  certain  piece  of  rail- 
road, wliat  will  be  the  expense  of  grading  15m.  6rur.  37rd.  ? 

Ans.  $87,781.33. 

^  48.   What   is   the   value  of  a  house-lot,  containing  40  square 
and  200  square  feet,  at  $  1.50  per  square  foot  ? 

Ans.  $  16635. 


REDUCTION.  109 

49.  How  many  yards  of  carpeting,  one  yard  in  width,  will  be 
required  to  carpet  a  room  18ft.  long  and  15ft.  wide  ? 

Ans.  30  yards. 

50.  A  certain  machine  will  cut  120  shingle-nails  in  a  minute, 
how  many  will  it  cut  in  47  days  7  hours,  admitting  the  machine 
to  be  in  operation  10  hours  per  day  ?  Ans.  3434400  nails. 

51.  In  a  field  80  rods  long  and  50  rods  wide,  how  many  square 
rods  ?     How  many  acres  ?  Ans.  25  acres. 

52.  How  long  will  it  take  to  count  18  millions,  counting  at  the 
rate  of  90  a  minute?  Ans.  138da.  21h.  20m. 

53.  A  merchant  purchased  9  bales  of  cloth,  each  containing  15 
pieces,  each  piece  23  yards,  at  8  cents  per  yard ;  what  was  the 
amount  paid  ?  Ans.  $  248.40. 

54.  Suppose  a  certain  township  is  6  miles  long  and  4^  miles 
wide,  how  many  lots  of  land  of  90  acres  each  does  it  contain  ? 

Ans.  192  lots. 

55.  The  pendulum  of  a  cei'tain  clock  vibrates  47  times  in  1 
minute;  how  many  times  will  it  vibrate  in  196  days  49m.  ? 

Ans.  13267583  times. 

56.  How  many  shingles  will  it  take  to  cover  a  roof,  each  of 
whose  equal  sides  is  36  feet  long,  with  rafters  16  feet  in  length, 
supposing  1  shingle  to  cover  27  square  inches  ? 

Ans.  6144  shingles. 

57.  How  many  times  will  the  large  wheels  of  an  engine  turn 
round  in  going  from  Boston  to  Portland,  a  distance  of  110  miles, 
supjiosing  the  wheels  to  be  12  feet  and  6  inches  in  circumfer- 
ence ?  Ans.  46464  times. 

58.  In  a  certain  house  there  are  25  rooms,  in  each  room  7 
bureaus,  in  each  bureau  5  drawers,  in  each  drawer  12  boxes,  in 
each  box  15  purses,  in  each  purse  178  sovereigns,  each  sovereign 
valued  at  $  4.84  ;  what  is  the  amount  of  the  money  ? 

Ans.  $135689400. 

59.  In  18rd.  5yd.  2ft.  llin.  how  many  inches? 

Ans.  3779  inches. 

60.  In  3779  inches  how  many  rods  ? 

Ans.  18rd.  5yd.  2ft.  llin. 

61.  Sold  5T.  17c%vt,  3qr.  181b.  of  potash  for  3  cents  per  pound; 
what  was  the  amount  ?  Ans.  $353.79. 

62.  A  gentleman  purchased  a  house-lot  that  was  25  rods  long 
and  16  rods  wide  for  $100,000,  and  sold  the  same  for  $1.25 
per  square  foot ;  what  did  he  gain  by  his  purchase  ? 

Ans.  $36,125. 
10 


110  COMPOUND   NUMBERS. 


ADDITION. 

101.  Addition  of  Compound  Numbers  is  the  process  of  finding 

the  amount  of  two  or  more   denominate  numbers  of  the   same 
kind,  when  one  or  more  of  them  is  compound. 


ENGLISH  MONEY. 

Ex.  1.  Paid  a  London  tailor  7£  13s.  6d.  2far.  for  a  coat; 
2£  17s.  9d.  Ifar.  for  a  vest;  3£  8s.  3d.  3far.  for  pantaloons; 
9£  lis.  8d.  3far.  for  a  surtout ;  what  was  the  amount  of  the 
bill?  Ans.  23£  lis.  4d.  Ifar. 

Having  written  units  of  the  same  de- 
nomination in  the  same  column,  we  find 
the  sum  of  the  farthings  to  be  9  farthings, 
equal  to  2d.  and  Ifar.  We  write  the 
It'ar.  under  the  tolumn  of  farthings^  and 
carry  the  2d.  to  the  column  of,  ppnce  ; 
the  sum  of  which  is  28d.,  equal  to  2s.  4d. 
Ans.  2  3      11      4      1         We  Avrite  the  4d.  under  the  tolumn   of 

pence,  and  carry  the  2s.  to  the  column  of 
shillings;  the  sum  of  which  is  51s.,  equal  to  2£  lis.  We  write  the 
lis.  under  the  column  of  sliillings,  and  carry  the  2£  to  tlie  column  of 
pounds  ;  and  have  lor  the  whole  amount,  23£  lis.  4d.  Ifar. 

The  same  result  can  be  arrived  at  by  reducing  the  numbers  as  they 
are  added  in  their  respective  columns.  Thus,  we  can,  beginning  with 
farthings,  add  in  this  way:  3far.  and  3far.  are  Gfar.,  equal  to  Id.  2iar., 
and  ifar.  are  id.  3far.,  and  2far.  are  Id.  51ar.,  ccjual  2d.  Ifar.  Writ- 
ing the  Ifar.  under  the  column  of  farthings,  carry  the  2d.  to  the  column 
of'^pence;  add  2d.  (carried)  and  8d.  are  lOd.,  and  3d.  are  13d.,  equal 
to  Is.  Id.,  and  9d.  are  Is.  lOd.,  and  Gd.  are  Is.  IGd.,  equal  to  2s.  4d. 
Writinij  the  4d.  under  the  column  of  pence,  carry  the  2s.  to  the  col- 
umn ofshillinrrs;  add  2s.  (carried)  and  lis.  are  13s.,  and  8s.  are  2l9., 
equal  to  l£  'is.,  and  17s.  are  l£  I8s.,  and  13s.  are  l£  31s.,  equal 
to  2£  lis.  AVriting  lis.  under  the  column  bf  sliillings,  carry  the 
2£  to  the  column  of  pounds,  and  so  find  the  whole  amount  to  be  as 
before. 

Thus  the  adding  of  compound  numbers  is  like  that  of  simple  num- 
bers, except  in  carrving  according  to  a  varying  scale  (Art.  82.  Note  ) 
A  like  difference  liolds  in  subtracting,  multiplying,  and  dividing  ot 
conipound  numbers. 


OPERATION. 

£            8.              d. 

far. 

7     13     6 

2 

2     17     9 

1 

3        8     3 

3 

9     118 

3 

101.  Whnt  !■!  nildilion  of  coinpoiinil  nuinhcvs  ?  IIow  do  you  nrrango 
coinpoinid  numbers  lor  iiddiiion '!  Why?  Wliat  is  the  ditfcrcncc  between 
addition  of  oompouud  and  addition  of  simple  numbers  i! 


ADDITION.  Ill 

Rule.  —  Write  all  the  given  numbers  so  that  units  of  the  same  denomi- 
nation may  stand  in  the  same  column. 

Add  as  in  addition  of  simple  numbers;  and  carry,  from  column  to 
column,  one  for  as  many  units  as  il  takes  of  the  denomination  added  to 
make  a  unit  of  the  denomination  next  higher. 

Proof.  —  The  proof  is  the  same  as  in  addition  of  simple 
numbers. 

Examples  for  Practice. 


TROY   WEIGHT. 

2. 

3. 

lb.         oz.       pwt. 

&r- 

lb. 

02.           pwt. 

gr. 

15     11     19 

22 

10 

10     10 

1  0 

71     10     13 

17 

81 

11     19 

23 

Q>5        9     17 

1  4 

47 

7        8 

1  9 

73     11     13 

13 

1  6 

9     10 

1  4 

'S^'    ' 

9 

33 

10        9 

21 

2  4  2-C)4     14 


APOTHECARIES'   WEIGHT. 


ft 

§ 

3 

9 

gr- 

81 

1  1 

6 

1 

1  9 

75 

10 

7 

2 

1  3 

1  4 

9 

7 

1 

1  2 

37 

8 

1 

1 

1  1 

G  1 

1  1 

3 

2 

3 

5. 

ft 

§ 

3 

B 

gr. 

35 

9 

G 

2 

19 

71 

1 

1 

1 

1  1 

37 

3 

3 

2 

1  2 

14 

4 

7 

1 

1  3 

fo 

■5 

G 

1 

1  7 

27243018 

AVOIRDUPOIS   WEIGHT. 
6.  7. 

T.        cwt.      qr.       lb.         oz.         dr.  T.        cwt.      qr.        lb.         oz.        dr. 

71     19     3     17     14     13         14     13     2     15     15     II 


14 

13 

1 

1  1 

13 

12 

13     17     3 

13 

1  1 

13 

39 

9 

3 

13 

9 

9 

4G     1  6     3 

1  1 

13 

1  0 

1  5 

17 

3 

1  6 

10 

1  4 

14     15     2 

7 

G 

0 

Gl 

1  G 

3 

13 

7 

8 

11     17     3 

10 

15 

1  1 

203 

17 

3 

23 

8 

8 

101. 

What 

is  the  rule 

?     The  proof  1 

112  COMPOUND   NUMBERS. 

CLOTH  MEASURE. 
8.. 


yd. 

qr. 

na. 

in. 

5 

3 

3 

2 

7 

1 

1 

2 

8 

3 

3 

1 

9 

1 

2 

2 

4 

3 

3 

2 

E.  E. 

qr. 

na. 

ia 

1  6 

3 

2 

1 

7.1 

1 

1 

2 

13 

3 

2 

1 

47 

3 

2 

2 

39 

2 

3 

2 

36      3       0       0 


LONG  MEASURE. 
10,  11- 


deg. 

m. 

fur. 

rd. 

ft. 

In. 

m. 

fur. 

rd. 

yd. 

ft. 

in. 

1  8 

19 

7 

15 

1  1 

1 

1  2 

7 

35 

5 

2 

1  1 

61 

47 

6 

39 

10 

1  1 

13 

6 

15 

3 

1 

10 

78 

32 

5 

1  4 

9 

9 

1  6 

1 

17 

1 

2 

5 

1  7 

59 

7 

36 

1  6 

10 

13 

4 

1  3 

2 

1 

9 

28 

56 

1 

30 

1  6 

1 

17 

7 

36 

5 

2 

7 

2  0  5        8:^     5     1  7     1  4^      8 

20^5        9       1     1*7     15        2 

SURVEYORS'   LIEASURE. 


12. 

13. 

m. 

fur. 

ch.  p. 

1. 

m. 

fur.  ch.  p.    1. 

1  7 

5 

8  3 

24 

14 

7  9  3  2  1 

1  G 

3 

7  1 

21 

• 

37 

10  3  16 

47 

7 

9  3 

19 

17 

7  8  3  17 

1  9 

6 

G  1 

1  6 

61 

6  5  3  16 

31 

7 

1  0 

20 

47 

1  1  0  23 

133 

7 

4  0 
14. 

0 

SQUARE 

MEASURE. 

15.  ■ 

A. 

R. 

p- 

ft. 

In. 

A.    K. 

p- 

yd.   ft.     in. 

67 

3 

39 

272 

143 

43  1 

1  5 

3  0   8     17 

78 

3 

1  4 

260 

1  1  6 

1  6  3 

3  9 

19  7  14  1 

14 

2 

31 

1  67 

1  35 

47  1 

1  6 

2  7  5    7  9 

6  7 

1 

17 

176 

131 

38  3 

17 

18  8    17 

49 

3 

3  1 

69 

1  17 

15  1 

3  2 

11  1  117 

278 

3 

15 

13  1i 

i 

66 
=  36 

278     3     15     131      102 


ADDITION.  113 

SOLID  MEASURE. 
16.  17. 


Tun. 

ft. 

in. 

1  7 

39 

1371 

6  1 

1  7 

17  11 

4  7 

1  G 

1  6G6 

7  1 

38 

1711 

47 

1  7 

1617 

Cord. 

ft. 

in. 

1  4 

1  1  G 

1169 

67 

1  1  3 

17  11 

9  0 

1  27 

9  69 

19 

98 

1376 

14 

37 

1414 

246     11    .1164 


WINE  MEASURE. 


J 

l«. 

19. 

Tun. 

hhd. 

gal. 

qt. 

pt. 

Tun. 

hhd. 

gal. 

qt. 

pt. 

01 

1 

62 

3 

1 

14 

3 

18 

3 

0 

7  1 

3 

1  4 

1 

1 

81 

1 

60 

3 

1 

60 

0 

17 

3 

0 

17 

3 

61 

3 

0 

1  4 

1 

51 

1 

1 

Gl 

3 

57 

3 

1 

57 

3 

1  4 

3 

1 

17 

1 

17 

1 

0 

265 

2 

35 

1 

0 

• 

BEER  MEASURE 

20. 

21. 

Tun. 

bbd. 

gal. 

qt. 

pt. 

Tun. 

hhd. 

gal. 

qt. 

pt. 

1  5 

3 

5  0 

3 

1 

•    67 

1 

51 

1 

0 

67 

3 

1  7 

3 

1 

15 

3 

1  6 

3 

1 

17 

1 

44 

1 

0 

44 

1 

45 

1 

1 

7  1 

3 

1  2 

3 

1 

15 

2 

1  2 

2 

1 

8  1 

1 

1  8 

1 

0 

67 

3 

35 

1 

0 

254 

1 

36 

0 

1 

DRY  MEASURE. 

22. 

23. 

ch. 

bu. 

pk. 

qt. 

pt. 

ch. 

bu. 

pk. 

qt. 

pt. 

15 

35 

3 

7 

1 

7  1 

17 

1 

1 

1 

1           CI 

1  6 

3 

G 

1 

1  G 

3  1 

3 

o 
O 

0 

5  1 

30 

1 

5 

0 

4  1 

1  4 

3 

1 

1 

4  2 

1  7 

2 

2 

1 

7  1 

1  7 

1 

0 

1 

1  4 

1  4 

1 

4 

1 

1  0 

1  0 

2 

3 

0 

186 

7 

1 

2 

0 

10* 

114  COMPOUND   NUMBERS. 

TIME. 


24. 

25. 

J' 

da.     h. 

m. 

8. 

w. 

da. 

h. 

m. 

8. 

57 

300  23 

59 

17 

15 

6 

23 

15 

17 

47 

169  15 

17 

38 

61 

5 

15 

27 

18 

29 

364  23 

42 

17 

71 

6 

21 

57 

58 

18 

178  16 

38 

47 

18 

5 

19 

39 

49 

49 

317  20 

52 

57 

87 

6 

19 

18 

57 

203     236     10     30     56 


CIRCULAR  MEASURE. 
26.  27. 


1 1 

28 

56 

58 

10 

2  1 

51 

37 

8 

13 

39 

57 

8 

19 

38 

49 

7 

1  7 

47 

48 

6 

1  7 

17 

18 

7 

09 

19 

5  1 

8 

18 

5  7 

45 

4 

1  7 

1  6 

39 

7 

27 

38 

48 

11     11     55     09 

Note.  —  The  sum 'of  the  sijrns,  •when  not  lesf?  than  12,  must  be  divided  by 
12,  and  only  the  remainder  be  written  down,  as  in  Ex.  26. 


SUBTRACTION. 

102.  Subtraclion  of  Compound  Niirahprs  is  tho  process  of  finding 

the  (lifFerence    between    two  denoiiiin:ite   numbers  of  the   same 
kind,  when  one  or  both  of  tiiem  are  compound. 

ENGLISH  MONEY. 

Ex.  1.  From  87£  9s.  6d.  3far.  take  52£  lis.  7d.  Ifar. 

Ilavinfj  plat'od  the  less  niunbor  under 
the  greater,  farthings  under  farthings, 
pence  under  pence,  etc.,  we  begin  with 
the  farthings,  thus  :  1  far.  iroui  3  far. 
leaves  2  far.,  which  we  set  under  the 
Rem.  3  4      17      11      2  colunui    of   farthings.      As  we   cannot 

102.  What  is  subtraction  of  compound  numbers?     How  do  you  arranjje 
the  numbers  for  subtraction  1 


OPERATION. 

£     B.     d. 

far. 

Min. 

87   9   6 

3 

Sub. 

52  11   7 

1 

SUBTRACTION. 


115 


take  7d.  from  6d.,  we  add  12d.  =  Is.  to  the  6d.,  making  18d.,  and 
then  subtract  the  7d.  from  it,  and  set  the  remainder,  Ud.,  under  the 
column  of  pence.  We  then  add  Is.  =  12d.  to  the  lis.  in  the  subtra- 
hend, making  Tis.,  to  compensate  for  the  12d.  we  added  to  the  bd.  in 
the  minuend.  (Art.  30.)  Again,  since  we  cannot  take  12s.  from  9s., 
we  add  2Us.  =  !£  to  the  9s.,  making  29s.,  from  which  we  take  the 
12s.,  and  set  the  remainder,  17s.,  under  the  column  of  shillings,  llav- 
ino-  added  l£  =  20s.  to  the  52£,  to  compensate  for  the  20s.  added  to 
the  9s.  in  the  minuend,  we  subtract  the  pounds,  and  obtain  34Jl  for 
the  remainder;  and  as  the  result  complete,  34£  17s.  lid.  2far. 

Rule.  —  Wi-ite  the  less  compound  number  under  the  greater^  so  that 
units  of  the  same  denomination  shall  stand  in  the  same  column. 

Subtract  as  in  subtraction  of  simple  nwnbers. 

If  any  number  in  the  subtrahend  is  larger  than  that  above  it,  add  to  the 
upper  number  as  many  units  as  make  one  of  the  next  higher  denomination 
before  subtracting,  and  carry  one  to  the  next  lower  number  before  subtract- 
ing it. 

Proof.  —  The  proof  is  the  same  as  in  subtraction  of  simple 
numbei's. 

Examples  for  Practice. 
2.  3. 


£ 

8. 

d.     far. 

78 

1    1 

5     2 

41 

13 

3     3 

3  6 

18 
4. 

1     3 

lb. 

oz. 

pwt.       gr. 

15 

3 

12     14 

9 

1  1 

17     2  1 

£ 

s. 

d.      far. 

765 

1  6 

10       1 

713 

1  7 

1  1       3 

TROY  WEIGHT. 


5". 


lb. 

711 
19 


1 
3 


pwt. 

o 
O 

18 


gr. 

17 
19 


ft 
15 
1  1 


3     14     17 

APOTHECARIES' 
6. 

§      3     9     gr. 

7     12     15 
9     7     119 


3     9     2     0     16 


WEIGHT. 

7. 

ft 

§     3 

9 

gr. 

1  61 

6     3 

1 

17 

97 

7     1 

2 

18 

102.  What  do  you  do  when  the  upper  number  is  smaller  than  the  lower  ? 
How  many  do  you  carry  to  the  next  denomination  i  What  is  the  rule  1 
The  proof'? 


116  COMPOUND   NUMBERS. 

AVOIRDUPOIS   WEIGHT. 
8.  9. 

T.  cwt.    qr.        lb.  oz.       dr.  T.        cwt.    qr.      lb.       oz.         dr. 

1171G1        5        014  11        101        113 

1917317        115  918311315 


97     18     1     12     14     15 


CLOTH  MEASURE. 


10. 

yd. 

qr.  na. 

In. 

1  5 

1   1 

2 

9 

3  3 

1 

11. 

E.  E. 

qr.  na. 

in. 

171 

2  2 

1 

19 

3  0 

2 

5     12     1 


LONG  JIEASURE. 


12. 

13. 

deg. 

m. 

fur. 

rd. 

yd. 

ft. 

in. 

deg. 

m. 

fur.  rd. 

ft. 

in. 

97 

3 

7 

31 

1 

1 

3 

13 

19 

1   1 

3 

1  9 

17 

1 

39 

1 

2 

7 

9 

28 

7  1 

1  6 

9 

7  7     5  5-J     5     3  1     4J     1     8 

) 


1=1     13     1       2     6 


77     55       7        5     1       12 


SURVEYORS'  MEASURE. 
14.  15. 

m.    fur.  cba.     p.         1.  m.     far.   cba.  p.        1. 

2135217  3171119 

958120  18173     23 


11     5     7     0     22 


SQUARE  MEASURE. 


16. 

17. 

A. 

R. 

p 

ft.      in. 

A. 

R. 

p.    yd. 

ft. 

In. 

1  1  6 

1 

1  3 

10  0  113 

13  9 

1 

17   18 

1 

30 

87 

3 

1  7 

2  0  0  117 

97 

3 

18  3  0 

1 

31 

28 

1 

35 

17  1^  .1  40 
i  =  o6 

28     1     35     172        32 


SUBTRACTION.  117 

SOLID  MEASURE. 
18.  19. 

T.  ft.  in.  Cords.        ft.  in. 

171301000  571181234 

98371234  199191279 


72  32  1494 

WINE  MEASURE. 

20.  21. 

T.         lihd.      gal.     qt.     pt.    gi.  T.     hhd.  gal.    qt.     pt.    gl. 

1713        8111  7111111 

99119313  933313 


72     1     51     1     1     2 

BEER  MEASURE. 

22.  23. 

T.     hhd.     gal.      qt.     pt.  T.      hhd.     gal.      qt.    pt. 

15     1     17     1     0  79     2        2     2     0 

931931  1931331 


5     1     51     1     1 


DRY  MEASURE. 

24.  25. 

ch.            bu.    pk.    qt.    pt.  ch.        bu.      pk.    qt.    pt. 

716        1210  7313301 

19        9311  1918131 


696     27     2     7     1 


TIME. 
26.  27. 


y- 

del. 

h. 

m. 

sec. 

V.   da. 

h. 

m. 

Bee. 

375 

15 

13 

17- 

5 

14  1 

3 

4 

15 

199 

137 

15 

1 

39 

9  6 

17 

37 

48 

175     243       4     15     26 

CIRCULAR  MEASURE. 
28.  29. 

8*  O  )  (I  S.  o  /  H 

11   7  13  15  1  23  37  39 

9291736  91538  47 


1   7  55  39  4   7  58  52 

Note.  —  In  Circular  Measure,  the  minuend  is  sometimes  less  than  the 
subtrahend,  as  in  Ex.  29,  in  which  case  it  must  be  increased  by  12  signs. 


118 


COMPOUND   NUMBERS. 


103.    To  find  the  difference  of  dates. 

Ex.  1.  What  is  the  difference  of  time  between  October  16th, 
1852,  and  August  9th,  1854  ?  Ans.  ly.  9mo.  23da. 


Min 
Sub. 

Rem. 


Min. 
Sub. 

Rem. 


FIRST  OPERATION. 

y.  mo.      da. 

1854        7        9 
1852        9     16 

1        9     23 

SECOND   OPERATION. 

1854        8        9 
1852     10     16 


Commencing  with  January,  the  first 
month  in  the  year,  and  counting  the 
months  and  days  in  the  later  date  up 
to  August  9th,  we  find  that  7mo.  and  9 
da.  have  elapsed ;  and  counting  the 
months  and  days  in  the  earlier  date,  up 
to  October  16th,  we  find  that  9mo.  and 
16da.  have  elapsed.  We,  therefore, 
write  the  numbers  for  subtraction  as  in 
the  first  operation.  The  same  rcsidt 
can  be  obtained  by  reckoning  the  72um- 
her  of  the  given  months  instead  of  the 
number   of  months   that   have   elapsed 

since  the  be<jinninnf  of  the  year,  and  writino;  the  numbers  as  in  the 

second  operation ;  —  written  either  way, 

The  earlier  date  being  placed  under  the  later,  is  subtracted,  as  by  the 
preceding  rule. 

Note.  —  In  finding:  the  difforcnce  between  two  dates,  and  in  computing 
interest  for  less  than  a  month,  30  days  are  considered  a  month.  In  lec/al  trans- 
actions, a  month  is  reckoned  from  any  day  in  one  month  to  the  coiTcsponding 
day  of  the  following  month,  if  it  has  a  corresponding  day,  otherwise  to 
its  end. 


9     23 


Examples  for  Pkactice. 


2.  What  is  the  time  from  March  21st,  1853,  to  Jan.  6th, 
1857  ?  Ans.  ^j.  9m.  15da. 

3.  A  note  was  given  Nov.  loth,  1852,  and  paid  April  25th, 
1857;  how  long  was  it  on  interest?  Ans.  4y.  5mo.  lOda. 

4.  Jolm  Quincy  Adams  was  born  at  Braintree,  Mass.,  July 
11th,  1707,  and  died  at  Washington,  D.  C,  Feb.  23,  1848;  to 
what  age  did  he  live?  Ans.  80y.  7mo.  12da. 

5.  Andrew  Jackson  was  born  at  Waxaw,  S.  C,  March  15th, 
1707,  and  died  at  Nashville,  Tenn.,  June  8th,   1845  ;  at  what 


age  did  he  die  ? 


Ans.  78}'.  2mo.  23da. 


10.3.  From  what  period  do  you  count  tlie  months  and  days  in  preparing 
dates  for  siihtraction  ?  How  do  you  arran'jc  the  dates  for  suhtractioii  ?  How 
subtract?  How  many  days  arc  considered  a  month  in  business  transactions? 
What  is  the  second  method  of  preparing  dates  for  subtraction  ? 


MISCELLANEOUS  EXERCISES.  119 


MISCELLANEOUS    EXERCISES, 

1.  What  is  the  amount  t)f  the  following  quantities  of  gold: 
41b.  8oz.  13pwt.  8gr.,  51b.  lloz.  19pwt.  23gr.,  81b.  Ooz.  17pwt. 
15gr.,  and  181b.  Ooz.  14pwt.  lOgr.? 

Ans.  ^71b.  7oz.  5pwt.  8gr. 

2.  An  apothecary  would  mix  71b3i  23  29  Igr.  of  rhubarb, 
21b  10  §  03  19  13gr.  of  caiitharides,  and  2tb  3i  7  3  29  17gr. 
of  opium  ;  what  is  the  weight  of  the  compound  ? 

Ans.  12ib5§  33  09  llgr. 

3.  Add  together  17T.  llcwt.  3qr.  111b.  12oz.,  IIT.  17cwt.  Iqr. 
191b.  lloz.,  53T.  19cwt.  Iqr.  171b.  8oz.,  27T.  19cwt.  3qr.  181b. 
9oz.,  and  IGT.  3cwt.  3qr.  01b.  13oz. 

Ans.  127T.  12cwt.  Iqr.  181b.  5oz. 

4.  A  merchant  owes  a  debt  in  London  amounting  to  7671£ ; 
what  remains  due  after  he  has  paid  1728£  17s.  9d.? 

Ans.  5942£  2s.  3d. 

5.  From  731b.  of  silver  there  were  made  261b.  lloz.  13pwt. 
14gr.  of  plate  ;  what  quantity  remained  ? 

Ans.  461b.  Ooz.  6pwt.  lOgr. 

6.  From  711b  8§  13  19  14gr.  take  71b  9 1  13  19  17gr. 

Ans.  631b  10 §  73  29  17gr. 

7.  From  28T.  13cwt.  take  lOT.  17cwt.  191b.  14oz. 

Ans.  17T.  15cwt.  3qr.  51b.  2oz. 

8.  A  merchant  has  3  pieces  of  cloth  ;  the  first  contains  37yd. 
3qr.  3na.,  the  second  18yd.  Iqr.  3na.,  and  the  third  31yd.  Iqr. 
2na. ;  what  is  the  whole  quantity  ?  Ans.  87yd.  3qr.  Ona. 

9.  Sold  3  loads  of  hay ;  the  first  weighed  2T.  13cwt.  Iqr. 
171b.,  the  second  3T.  171b.,  and  the  third  IT.  3qr.  lUb.  ;  what 
did  they  all  weigh  ?  Ans.  6T.  14cwt.  Iqr.  201b. 

10.  What  is  the  sum  of  the  following  distances:  16ra.  7fur. 
18rd.  14ft.  llin.,  19m.  Ifur.  13rd.  16ft.  9in.,  97m.  3fur.  27rd. 
13ft.  3in.,  and  47m.  5fur.  37rd.  ISft.  lOin.  ? 

Ans.  181m.  2fur.  18rd.  9ft.  Sin. 

11.  From  76yd.  take  18yd.  3qr.  2iia.     Ans.  57yd.  Oqr.  2na. 

12.  From  20m.  take  3m.  4fur.  18rd.  13ft.  Sin. 

Ans.  16m  3fur.  21rd.  2ft.  lOin. 

13.  From  144A.  3R.  take  18A.  IR.  17p.  200ft.  lOOin. 

Ans.  126A.  IR.  22p.  71ft.  80in. 


120  COMPOUKD   ^' UMBERS. 

14  From  18  cords  take  3  cords  100ft.  lOOOln. 

Ans.  14  cords  27ft.  728:n. 

15.  A  gentleman  has  three  farms;  the  first  contains  1G9A 
3R.  lop.  227ft.,  the  second  187A.  IR.  l;3p.  IGoft.,  and  the  third 
217A.  2R.  28p.  IGjft. ;  what  is  the  whole  quantity? 

Ans.  574A.  3R.  20p.  12irt. 

16.  There  are  3  piles  of  wood ;  the  first  contains  18  cords 
116ft.  lOOOin.,  the  second  17  cords  111ft.  1600in.,  and  the  third 
21  cords  lOUft.  1716in.  ;  how  much  in  all? 

Ans.  58  cords  82ft.  8G0in. 

17.  From  17T.  take  5T.  18ft.  765in.    Ans.  IIT.  21ft.  963in. 

18.  From  169gaL  take  76gal.  3qt.  Ipt. 

Ans.  92gal.  Oqt.  Ipt. 

19.  From  17ch.  18bu.  take  5ch.  20bii.  Ipk.  7qt. 

Ans.  llch.  33bu.  2pk.  Iqt. 

20.  From  837.  take  A7j.  lOmp.  27d.  18h.  50m.  14s. 

Ans.  35y.  Imo.  2d.  5h.  9m.  46s. 

21.  From  llS.  15°  36'  15"  take  5S.  18°  50'  18". 

Ans.  5S.  26°  45'  57". 

22.  John  Thomson  has  4  casks  of  molasses  ;  the  first  con- 
tains lG7gal.  3qt.  Ipt.,  the  second  186gal.  Iqt.  Ipt.,  the  third 
lOSgal.  2qt.  Ipt.,  and  the  fourth  123gal.  3qt.  Opt.;  how  much  is 
the  whole  quantity  ?  Ans.  58Ggal.  2qt.  Ipt. 

23.  Add  together  17bu.  Ipk.  7qt.  Ipt.,  18bu.  3pk.  2qt.,  19bu. 
Ipk.  3qt.  Ipt.,  and  51bu.  3pk.  Oqt.  Ipt. 

Ans.  107bu.  Ipk.  5qt.  Ipt. 

24.  James  is  13y.  4mo.  13J.  old,  Samuel  is  12y.  llmo.  23d., 
and  Daniel  is  18y.  9mo.  29d. ;  what  is  the  sum  of  their  united 
ages?  Ans.  45y.  2mo.  od. 

25.  Add  together  ISj.  34od.  13h.  37m.  15?..  87y.  lG9d.  12h. 
16m.  28s.,  3l6y.  144d.  20h.  53m.  18s.,  and  13y.  366d.  21h.  57m. 
15s.  Ans.  436y.  290d.  20h.  44m.  I63. 

26.  A  carpenter  sent  two  of  his  apprentices  to  ascertain  the 
length  of  a  certain  fence.  The  first  stated  that  it  was  I7rd.  16ft. 
llin.,  the  second  said  it  was  18rd.  Sin.  The  carpenter  finding  a 
discrepancy  in  their  statements,  and  fearing  they  might  both  be 
wrong,  ascertained  the  true  length  himself,  which  was  17rd.  5yd. 
1ft.  llin. ;  how  much  did  each  d'lfCcr  from  the  other? 

27.  From  a  mass  of  silver  weighing  1061b.,  a  goldsmith  made 
36  spoons,  weighing  51b.  1  loz.  12pwt.  l.")gr.  ;  a  tankard,  31b.  Ooz. 
13pwt.  14gr. ;  a  vase,  71b.  lloz.  14pwt.  23 "t.  ;  how  much  uu- 
wrought  silver  remains?  Ans.  881b.  lloz.  18pwt.  20gr. 


MULTIPLICATION.  121 

28.  From  a  piece  of  cloth,  containing  17yd.  3qr.,  there  were 
taken  two  garments,  the  first  measuring  3yd.  3qr.  2na.,  the 
second  4yd.  Iqr.  3na. ;  how  much  remained  .'' 

Ans.  9yd.  Iqr.  3na. 

29.  Venus  is  3S.  18"  45'  15''  east  of  the  Sun,  Mars  is  7S.  15' 
3G'  18"  east  of  Venus,  and  Jupiter  is  5S.  21°  38'  27"  east  of 
Mars  ;  how  far  is  Jupiter  east  of  the  Sun  ?  Ans.  4S.  26°. 

30.  The  longitude  of  a  certain  star  is  3S.  18°  14'  35",  and 
the  longitude  of  Jupiter  is  llS.  25°  30'  50";  how  far  will  Ju- 
piter have  to  move  in  his  orbit  to  be  in  the  same  longitude  with 
the  star  ?  Ans.  3S.  22°  43'  45". 


MULTIPLICATION. 

104.  Multiplication  of  Componnd  Nnmbers  is  the  process  of  taking 
a  compound  number  any  proposed  number  of  times. 

105.  When  the  multiplier  is  12  or  less. 

Ex.  1.   If  an  acre  of  land  cost  14£  53.  8d.  2far.,  what  will 
9  acres  cost  ?  Ans.  128£  lis.  4d.  2far. 

OPERATION.  We  write  the  multiplier  under 

-^         ,.               ^         «.     d.    far.  the   lowest   denomination    of   the 

Multiplicand      14       5     8     2  multiplicand,  and  then  say  0  times 

Multiplier      9  2far.  are  18far.,  equal  to  4d.  and 

Product  12  8    11      4     9     H^^^'     ^^^  '^'''^*^  ^^^   2far.  under 

rroduct  i  Z  »    1  i      4     J     thg  ^yj^jjgj.  ^uitipii^^j^  reserving 

the  4d.  to  be  added  to  the  next 
product.  We  then  say  9  times  8d.  are  72d.,  and  the  4d.  make  76d., 
equal  to  6s.  and  4d.,  and  write  the  4d.  under  the  column  of  pence, 
reserving  the  6s.  to  be  added  to  the  next  product.  Then,  9  times 
5s.  are  45s.,  and  6s.  make  51s.,  equal  to  2£  and  lis.,  and  write  the 
lis.  under  the  column  of  shillings,  reserving  the  2£  to  be  added  to 
the  next  product.  Again,  9  times  14£  are  126£,  and  2£  make 
128£,  which  we  write  under  the  column  of  pounds;  and  have  128£ 
lis.  4d.  2far.  for  the  answer. 


104    "What  is  multiplication  of  compounfl  numbers?  — 105.  Explain  the 
operation.  By  what  do  you  divide  the  product  of  each  denomination  1   What 
do  you  do  with  the  quotient  and  remainders  thus  obtained  ? 
11 


122  COMPOUND   NUMBERS. 

Rule.  —  Multiply  each  denomination  of  the  compound  numher  as  in 
multiplication  of  simple  numbers,  and  carry  as  in  addition  of  compound 
numbers. 

Note.  —  Going  a  second  time  carefully  over  the  work  is  a  good  way  of 
testing  its  accuracy.  On  learning  Division  of  Compound  Numbers,  the  pupil 
will  lind  by  that  rule  a  better  method  of  proving  multiplication  of  compound 
numbers. 


Examples 

FOR  Practice. 

2. 

3. 

4. 

5. 

5 

B.    d. 

6    8 
2 

£ 

19 

8. 
11 

d 

7 
3 

£                B. 

25     17 

d. 
1  1 

5 

£                 8.                d. 

18     15     8f 
6 

10  13     4 

i 

58 

14 

9 

129        9 

7 

112    14     4  j 

6. 

7. 

8. 

cwt. 

18 

qr.        lb. 
3       17 

2        5 

oz. 

10 
6 

12 

Ton. 

14 
103 

cwt.     qr.       lb. 

15     3     12 

7 

110       9 

cwt.  ^qr.         lb.    oz. 

19     1        8    15 
8 

113 

154    2     21       8 

9. 

* 

10. 

11. 

lb. 

15 

oz. 

14 

dr. 

13 

9 

m. 

97 

fur. 
7 

rd.          ft. 

14     13 
6 

deg. 

18 

m.      fur.      rd. 

12     6     18 

8 

143 

5 
12. 

5 

587 

4 

8^12 

145 

3  3     2     1  Of 
13. 

rd. 

23 

yd.     ft. 

3     2 

In. 
9 

9 

fur. 

9 

rd.         ft.         in.' 

31     16     11 
IQ 

213209  98042 

NoTR.  —  The  answers  to  the  following  questions  are  found  in  the  cor- 
responding questions  in  Division  of  Compound  Numbers,  p.  126. 

14.  What  cost  7  yards  of  cloth  at  18s.  9d.  per  yard  ? 

15.  If  a  man  travel  12m.  3fur.  20rd.  in  one  day,  bow  far  will 
he  travel  in  9  days  ? 

IG.  If  1  acre  produce  2  tons  13cwt,  191b.  of  hay,  what  will 
8  acres  produce? 

»  — " 

104.  What  is  the  rule  ?    How  may  the  work  be  tested  1 


MULTIPLICATION.  123 

17.  If  a  family  consume  49gal.  3qt.  Ipt.  of  molasses  in  one 
month,  what  quantity  will  be  sutHcient  for  one  year? 

18.  John   Smith   has    12    silver    spoons,    each   weighing  3oz. 
17pwt.  14gr. ;  what  is  the  weight  of  all  ? 

19.  Samuel  Johnson  bought  7  loads  of  timber,  each  measur- 
ing 7  tons  37ft. ;  what  was  the  whole  quantity  ? 

20.  If  the  moon  move  in  her  orbit  13°  11'  35"  in  1  day,  how 
far  will  she  move  in  10  days  ? 

21.  If  1  dollar  will  purchase  21b  8  §  73  19  lOgr.  of  ipecacu- 
anha, what  quantity  would  9  dollars  buy  ? 

22.  If  1  dollar  will  buy  2A.  3R.  15p.  30yd.  8ft.  lOOin.  of 
wild  land,  what  quantity  may  be  purchased  for  12  dollars? 

23.  Joseph  Doe  will  cut  2  cords  97ft.  of  wood  in  1  day ;  how 
much  will  he  cut  in  9  days  ? 

24.  If  1  acre  of  land  produce  3ch.  6bu.  2pk.  7qt.  Ipt.  of  corn, 
what  will  8  acres  produce  ? 

106i   When  the  multiplier  is  a  composite  number,  and 
none  of  its  factors  exceed  12. 

Ex.  1.   What  cost  24  yards  of  broadcloth  at  2£  7s.  lid.  per 
yard  ?  Ans.  57£  lOs.  Od. 

OPERATION. 

^  8.  d.  ^  24  is  equal  to  4  X  6 ;  we 
2  7  11=  price  of  1  ^rd.  therefore  multiply  the  price 
4  ^  of  1  yard  by  4,  and  obtain 
-  —  -  '  p  ,  ,  *^<^  price  of  4  yards,  which 
y  1  1  8  =  price  of  4  yards.  we  multiply  by  6,  and  obtain 
6  the  price  of  24  yards. 


5  7     10        0  =  price  of  24  yards. 

Ex.  2.   What  cost  360  tons  of  iron  at  17£  16s.  Id.  per  ton  ? 

Ans.  6409£  10s.  Od. 

OPERATION. 

£  s.         d. 

17     16       1  =  price  of  1  ton.  360  is  equal  to  6  x  6  X 

6  10.     We  therefore  multiply 

^  n  a      A  n        c             •         /»  «  ,  ^^^  price  of  1  ton  by  6,  and 

10  6     16        6  =  price  of  6  tons,  obtain  the  price  of  6  tons, 

^  which  multiplied  by  6  gives 

6  4  0     19       0  =  price  of  36  tons.    '^\  Pf'^^  f  ?^  *°"^'  ^."^ 
,  /^         ^  that  by  10  gives  the  price 

- tJL  of  3G0  tons. 

6409     10       0  =  price  of  360  tons. 


124 


COMPOUND   NUMBERS. 


Rule.  —  Multiply  hy  the  factors  of  the  composite  number  in  succes- 
sion. 

Examples  for  Practice. 

3.  If  a  man  travel  3in.  7fur.  18rd.  in  one  day,  how  far  -would 
he  travel  in  30  days  ? 

4.  If  a  load  of  hay  weigh  2  tons  7cwt.  3qr.  181b.,  what  would 
be  the  weight  of  84  similar  loads .'' 

5.  When  it  requires  7yd.  3qr.  2na.  of  silk  to  make  a  lady's 
dress,  what  quantity  would  be  sufficient  to  make  72  similar 
dresses  ? 

6.  A  tailor  has  an  order  from  the  navy  agent  to  make  132 
garments  for  seamen  ;  how  much  cloth  wUl  it  take,  supposing 
each  garment  to  require  3yd.  2qr.  Ina.  ? 

# 

107.  When  the  multiplier  is  not  a  composite  number, 
and  exceeds  12,  or,  if  a  composite  number,  and  any  of  its 
factors  exceed  12. 

Ex.  1.  What  cost  379cwt.  of  iron  at  3£  16s.  8d.  per.  cwt.  ? 

Ans.  14o2£  16s.  8d. 


OPEKATION. 
£  8.  d. 

3     16      8  =  cost  of  IcAvt. 
10 


38 


6      8 
10 


383 


6 


8 
3 


cost  of  lOcwt. 
cost  of  lOOcwt. 


115  0        0      0  =  cost  of  300cwt. 
2  6  8        6      8  =  cost  of  70cwt. 
3  4     10      0  =  cost  of  9cwt. 


1452     16      8  =  cost  of  379cwt. 
the  several  products,  we  obtain  1452£ 


Since  379  is  not  a  com" 
posite  number,  we  cannot 
resolve  it  into  factors ;  but 
we  may  separate  it  into 
parts  ;  thus,  379  =  300  + 
70  -|-  9-  In  the  operation 
we  first  multiply  by  10,  and 
then  by  10,  to  get  the  cost 
of  lOOcwt.  To  find  the  cost 
of  SOOowt.,  we  multiply  the 
cost  of  1  OOcwt.  by  3  ;  and  to 
find  the  cost  of  70cwt.,  we 
multiply  the  cost  of  lOowt. 
by  7  ;  and  then,  to  find  the 
cost  of  9cwt.,  we  multiply  the 
cost  of  Icwt.  by  9.  Adding 
IGs.  8d.  for  the  answer. 


Rule.  —  Having  resolved  the  multiplier  into  any  convenient  parts,  as 
of  units,  tens,  etc.,  multiply  by  these  several  parts,  adding  together  the 
producUt  thus  obtained  for  the  required  result. 

106.  What  is  the  rule  for  mnltiplyinjr  by  a  composite  number?  Reason 
for  the  rule?  —  107.  How  flo  you  find  the"  cost  of  SOOcwt.  in  the  example? 
0f  70cwt.  ?  Of  9cwt.  ?  Wliat  is  the  rule  when  the  multiplier  is  large,  and 
is  not  a  composite  number? 


DIVISION.  125 

Examples  fob  Practice. 

2.  If  1  dollar  will  buy  171b.  lOoz.  13dr.  of  beef,  how  much 
may  be  bought  for  62  dollars  ? 

3.  What  cost  97  tons  of  lead  at  2£  17s.  9^d.  per  ton  ? 

4.  If  a  man  travel  17m.  3fur.  19rd.  3yd.  2ft.  7in.  in  one  day, 
how  far  would  he  travel  in  38  days  ? 

5.  If  1  acre  will  produce  27bu.  3pk.  6qt.  Ipt.  of  corn,  what 
will  98  acres  produce  ? 

6.  If  it  require  7yd.  3qr.  2na.  to  make  1  cloak,  what  quantity 
would  it  require  to  make  347  cloaks  ? 

7.  One  ton  of  iron  will  buy  13A.  3R.  14p.  18yd.  7ft.  76in.  of 
land  ;  how  many  acres  wiU  19  tons  buy  ? 

8.  If  1  ton  of  copper  ore  will  purchase  17T.  14cwt.  3qr.  181b. 
14oz.  of  iron  ore,  how  much  can  be  purchased  for  451  tons  ? 

Ans.  8003T.  17cwt.  Iqr.  121b.  lOoz. 


DIVISION 


£ 


108.  Division  of  Compound  Numbers  is  the  process  of  dividmg  a 

compound  number  into  any  proposed  number  of  equal  parts. 

109.  To  divide  when  the  divisor  does  not  exceed  12. 

Ex.  1.  If  9  acres  of  land  cost  128£  lis.  4d.  2far.,  what  is  the 
value  of  1  acre  ?  Ans.  14£  5s.  8d.  2far. 

OPERATION.  Havinst  divided  the  128£  by  9,  we  find 

p.        d.     far.        the  quotient  to  be  14£  and   2£  remaininjr. 

9)128    11      4     2  We  place  the  14£   under   the  128£,  and 

-  reduce  2£  to   shillings,  making   40s.,  and 

14        5      o     z  adding  the  lis.  in  the  dividend,  we  have 

51s      We  next  divide  the  51s.  by  9,  and 

write  the  quotient  5s.  under  the  lis.,  and  to  the  remainder  6s.,  equal 

to  72d.,  add  the  4d.,  making  76d.     The  76d.  we  divide  by  9,  and  write 

the  quotient  8d.  under  the  4d.,  and  to  the  remainder  4d  ,  equal  to 

108.  What  is  division  of  compound  numbers  ?  — 109.  Where  do  you  begin 
to  divide  ■?     Why  ?     When  there  is  a  remainder  after  dividing  any  one  de- 
nomination, what  must  be  done  with  it  1 
11* 


126  COMPOUND   NUMBERS. 

16far.,  we  add  the  2far.,  making  ISfar.  The  18far.  we  divide  by  9, 
and  obtain  2far.  for  a  quotient,  which  we  place  under  the  2far.  in  the 
dividend ;  and  thus  find  the  answer  to  be  14£  5s.  8d.  2far. 

RuLK.  —  Divide  as  in  division  of  simple  numbers,  each  denomination 
in  its  order,  beginning  with  the  highest. 

If  there  be  a  remainder,  reduce  it  to  the  next  lower  denomination,  add' 
ing  in  the  number  already  of  this  denomination,  if  any,  and  divide  as 
before. 

Proof.  —  The  same  as  in  simple  numbers. 

Note.  —  When  the  divisor  and  dividend  are  denominate  numbers,  and 
one  or  both  are  compound,  reduce  them  to  the  same  denomination,  and  then 
proceed  as  with  simple  numbers. 

2.  3.  4. 

£  8.         d.  £  8.        d.  £  e.  d. 

2)10     13     4  3)58     149  5)129       9        7 

5       6    8  19     117  25     17     11 

5.  6.  7. 

£  B.      d.  far.  cwt.     qr.        lb.    oz.  ton.      cwt.    qr.      lb. 

6  )1  12    14   4   2     6 ) 1 13    2      5    12     7)103110      9 

18  15  8   3  18   3   17   10  14   15   3   12 

8.  9.  10. 

cwt.      qr.     lb.        oz.  lb.  oz.       dr.  m.       fur.  rd.    ft. 

8)154221      8       9)143      5      5       6)5874812 

19  1      8   15  15   14   13 

11.  12.  13. 

deg.        m.    fur.     rd.  rd.      yd.    ft.  in.  fur.    rd.  ft.  in. 

8  )  1  4  5   3  3   2    1  Of        9  )213    2   0   9         1  0  )98   0   4   2 

Note.  —  The  answers  to  the  following  questions  arc  found  in  the  corre- 
sponding numbers  in  Multiplication  of  Compound  Numbers. 

14.  What  costs  1  yard  of  cloth,  when  7yd.  can  be  bought  for 
6£  lis.  3d.? 

15.  If  a  man,  in  9  days,  travel  112m.  Ifur.  21rd.,  how  far  will 
he  travel  in  1  day  ? 

16.  If  8 'acres  produce  21T.  5cwt.  2qr.  21b.  of  hay,  what  will 
1  acre  produce? 

109.  What  is  the  rule  for  division  of  compound  numbers  ? 


DIVISION.  127 

17.  If  a  family  consume  in  1  year  598gal.  2qt.  of  molasses, 
how  much  will  be  necessary  for  1  month  ? 

18.  John   Smith  has    12  silver  spoons,  weighing   31b.   lOoz. 
llpwt. ;  what  is  the  weight  of  each  spoon  ? 

19.  Samuel  Johnson  bought  7  loads  of  timber,  measuring  55T. 
19ft. ;  what  was  the  quantity  in  each  load  ? 

20.  If  the  moon,  in  10  days,  move  in  her  orbit  4S.  11°  55'  50", 
how  far  does  she  move  in  1  day  ? 

21.  K  $9  will  buy  241b  8§    33    IB  lOgr.  of  ipecacuanha, 
how  large  a  quantity  will  $  1  purchase  ? 

22.  When  $  12  will  buy  34 A.  OR.  32p.  8yd.  5ft.  48in.  of  wild 
land,  how  mucli  will  $  1  buy  ? 

23.  Joseph  Doe  will  cut  24  cords  105  feet  of  wood  in  9  days ; 
how  much  will  he  cut  in  1  day  ? 

24.  When  8  acres  of  land  produce  25ch.  17bu.  3pk.  4qt.  of 
grain,  what  will  1  acre  produce  ? 

110.     When  the  divisor  is  a  composite  number,  and 
none  of  its  factors  exceed  12. 

Ex.   1.  When  24  yards  of  broadcloth  are  sold  for  57£  10s. 
Od.,  what  is  the  price  of  1  yard?  Ans.  2£  7s.  lid. 


24   is  equal  to  6   X  4. 


OPERATION. 

6  )  5^7     10      0  =  price  of  24  yards.        ^^   therefore   divide  the 

i ^  •'  price  by  one  of  these  lac- 

4)9     11       8  =  price  of    4  yards.        tors,  and  the  quotient  aris- 

o         7    11  •        f    1  1  iug  by  the  other. 

.2        7   11  =  price  or    1  yard. 

KuLE.  —  Divide  by  the  factors  of  the  composite  number  in  succession. 

Examples   for   Practice. 

2.  If  360  tons  of  iron  cost  6409£  10s.  Od.,  what  is  the  cost  of 
1  ton? 

3.  If  a  man  travel  117m.  7fur.  20rd.  in  30  days,  how  far  will 
he  travel  in  1  day  ? 

4.  If  84  loads  of  hay  weigh  201  tons  6cwt.  Oqr.  121b.,  what 
will  1  load  w  eigh  ? 

5.  When  72  ladies  require  567yd.  Oqr.  Ona.  for  their  dresses, 
how  many  yards  will  be  necessary  for  one  lady  ? 

110.  How  docs  it  appear  that  dividin;^  ])y  q  in  Ex.  1  gives  the  price  of  4 
yards  ?     How  do  you  divide  by  a  composite  number'? 


128  COMPOUND   NUMBERS. 

6.  When  132  sailors  require  470yd.  Iqr.  of  cloth  to  make  their 
garments,  how  many  yards  will  be  necessary  for  1  sailor  ? 

111.  When  the  divisor  is  not  a  composite  number,  and 
exceeds  12,  or,  if  a  composite  number,  and  any  of  its 
factors  exceed  12. 

Ex.  1.  If  23cwt.  of  iron  cost  171£  Is.  3d.,  what  cost  Icwt.  ? 

Ans.  7£  8s.  9d. 

OPEEATION. 

2  3  )  1  7  1    1   3  (  7£ 

161  We  divide  the  pounds  by  23,  and  obtain 

r~^  7£    for   the   quotient,   and   10£    remaining, 

^  "which  we  reduce  to  shillings,  and  add  the 

^  ^  Is.,  and  again  divide  by  23,  and  obtain  8s. 

2  3  ^  2  0  1  (^  8s.  ^^^  ^^^^  quotient.     The  remainder,  1 7s.,  we 

'  ,  Q  J          *  reduce  to  pence,  and  add  the  3d.,  and  again 

divide   by   23,  and  obtain   9d.  for  the  quo- 

1  7  tient ;  and,  by  uniting  the  several  quotients, 

12  '^^  obtain  7£  8s.  9d.  for  the  answer.    There^ 

fore, 

2  3  )  2  0  7  (  9d. 

207 

The  method  of  operation  is  like  that  by  the  general  rule  (Art.  109), 
excepting  more  of  the  work  is  written  down. 

2.  If  $  62  will  buy  10951b.  14oz.  6dr.  of  beef,  how  much  may 
be  obtained  for  $  1  ? 

3.  Paid  280£  5s.  9^d.  for  97  tons  of  lead ;  what  did  it  cost 
per  ton  ? 

4.  If  a  man  travel  662m.  4fur.  28rd.  3yd.  2ft.  2in.  in  38  days, 
how  far  will  he  travel  in  1  day  ? 

5.  When  98  acres  produce  2739bu.  Ipk.  5qt.  of  grain,  what 
will  1  acre  produce  ? 

6.  A  tailor  made  347  garments  from  2732yd.  2qr.  2na.  of 
cloth ;  what  quantity  did  it  take  to  make  1  garment  ? 

7.  When  19  tons  of  iron  will  purchase  262A.  3R.  37p.  25yd. 
1ft.  40in.  of  land,  how  much  may  be  obtained  for  1  ton  ? 

8.  451  tons  of  copper  ore  will  purchase  8003T.  17cwt.  Iqr. 
121b.  lOoz.  of  iron  ore,  how  much  will  1  ton  purchase  ? 

Ans.  17T.  14cwt.  3qr.  181b.  14oz. 

III.  When  the  divisor  is  large,  and  not  a  composite  number,  how  is  tho 
division  performed  1 


MSCELLANEOUS  EXAMPLES.  129 


MISCELLANEOUS    EXAMPLES. 

1.  Bought  30  boxes  of  sugar,  each  containing  8cwt.  3qr.  201b., 
but  having  lost  68cwt.  2qr.  01b.,  I  sold  the  remainder  for  1£  17s. 
6d.  per  cwt. ;  what  sum  did  I  receive  ?  Ans.  37o£. 

2.  A  company  of  144  persons  purchased  a  tract  of  land  con- 
tainhig  11067A.  IR.  8p.  John  Smith,  who  Avas  one  of  the  com- 
pany, and  owned  an  equal  share  with  the  others,  sold  his  part 
of  the  laud  for  Is.  9^d.  per  square  i*od ;  what  sum  did  he  re- 
ceive ?  Ans.  1101£  12s.  l^d. 

3.  The  exact  distance  from  Boston  to  the  mouth  of  the  Colum- 
bia River  is  2644m.  3fur.  12rd.  A  man,  starting  from  Boston, 
traveled  100  days,  going  18m.  7fur.  32rd.  each  day ;  required 
his  distance  from  the  mouth  of  the  Columbia  at  the  end  of  that 
time.  Ans.  746m.  7fur.  12rd. 

4.  James  Bent  was  born  July  4,  1798,  at  3h.  17m.  A.  M. ; 
how  long  had  he  hved  Sept.  9,  1807,  at  llh.  19m.  P.M., 
reckoning  365  days  for  each  year,  excepting  the  leap  year  1804, 
which  has  366  days  ?  Ans.  3353da.  20h.  2m. 

5.  The  distance  from  Vera  Cruz,  in  a  straight  hne,  to  the  city 
of  Mexico,  is  121m.  5fur.  If  a  man  set  out  from  Vera  Cruz  to 
travel  this  distance,  on  the  first  day  of  January,  1848,  which  was 
Saturday,  and  traveled  3124rd.  per  day  until  the  eleventh  day  of 
January,  omitting,  however,  as  in  duty  bound,  to  travel  on  the 
Lord's  day,  how  far  would  he  be  from  the  city  of  Mexico  on  the 
morning  of  that  day  ?  Ans.  43m.  4fur.  8rd. 

6.  Bought  16  casks  of  pota?;h,  each  containing  7cwt.  3qr. 
181b.,  at  5  cents  per  pound.  I  disposed  of  9  casks  at  6  cents 
per  pound,  and  sold  the  remainder  at  7  cents  per  pound  ;  what 
did  I  gain  ?  Ans.  $  182.39. 

7.  A  mei'chant  purchased  in  London  17  bales  of  cloth  for 
17£  ISs.  lOd.  per  bale.  He  disposed  of  the  cloth  at  Havana  for 
sugar  at  l£  17s.  6d.  per  cwt.  Now,  if  he  purchased  144cwt.  of 
sugar,  what  balance  did  he  receive  ?  Ans.  35£  Os.  2d. 

8.  A  and  B  commenced  traveling,  the  same  way,  round  an 
island  50  miles  in  circumference.  A  travels  17m.  4fiu-.  30rd. 
a  day,  and  B  travels  12m.  3fur.  20rd.  a  day  ;  required  how 
far  they  are  apart  at  the  end  of  10  days. 

Ans.  Im.  4fur.  20rd. 


130  PROPERTIES   OF   NUMBERS. 

9.  Bought  760  barrels  of  flour  at  $  5.75  per  barrel,  which  I 
paid  for  in  iron  at  2  cents  per  pound.  The  purchaser  afterwards 
sold  one  half  of  the  iion  to  an  ax  manufacturer ;  what  quantity 
did  he  sell?  Ans.  54T.  12cwt.  2qr. 

10.  Bought  17  house-lota,  each  containing  44  perches,  200 
square  feet.  From  this  purchase  I  sold  2A.  2R.  240ft.,  and  the 
remaining  quantity  I  disposed  of  at  Is.  2^d.  per  square  foot ; 
what  amount  did  I  receive  for  the  last  sale  ? 

Ans.  5914£  19s.  oid. 

11.  J.  Spofford's  farm  is  100  rods  square.  From  this  he  sold 
H.  Spaulding  a  fine  house-lot  and  garden,  containing  5 A.  3R. 
17p.,  and  to  D.  Fifts  a  farm  oOrd.  square,  and  to  R.  Thornton  a 
farm  containing  3000  square  rods ;  what  is  the  value  of  the  re- 
mainder, at  $  1.75  per  square  rod?  Ans.  $  6235.25. 

12.  Bought  78A.  3R.  30p.  of  land  for  $  7000,  and,  having  sold 
10  house-lots,  each  30rd.  square,  for  $  8.50  per  square  rod,  I  dis- 
pose of  the  remainder  for  2  cents  per  square  foot.  How  much 
do  I  gain  by  my  bargain  ?  Ans.  $  89265.35. 


PROPERTIES    OF    NUMBERS. 

112i    An  Integer  is  a  whole  number ;  as  1,  6,  13. 

All  numbers  are  either  odd  or  even. 

An  Odd  Number  is  a  number  that  cannot  be  divided  by  2  with- 
out a  remainder  ;  thus,  3,  7,  11. 

An  Even  Number  is  a  number  that  can  be  divided  by  2  without 
a  remainder;  thus,  4  8,  12. 

Integers  are  also  either  jonme  or  composite  numbers. 

A  Prime  Number  is  a  number  which  can  be  exactly  divided  by 
no  integer  excei)t  itself  or  1  ;  as,  1,  8,  5,  7. 

A  Composite  Number  is  a  number  which  can  be  exactly  divided 
by  an  integer  other  than  itself  or  1  ;  as,  6,  9,  14. 

Numbers  are  prime  to  each  other  when  they  have  no  factor 
(Art.  4 1 )  in  common ;  thus,  7  and  1 1  ai*e  prime  to  each  other,  as 
are  also  4,  15,  imd  19. 

112.  What  is  an  integer  ?  What  are  all  integers  ?  What  is  an  odd  num- 
ber ?  An  even  number  7  Wliat  other  distinctions  of  nunihers  are  men- 
tioned 1  What  is  a  prime  number  1  When  are  numbers  prime  to  each 
other  1     What  is  a  composite  number  1  * 


PROPERTIES   OF   NUiMBERS. 


131 


All  the  prime  numbers  not  larger  than  1109  are  included  in 


the  following 


TABLE   OF  PRIME  NUMBERS. 


1 

59 

139 

233 

337 

439 

557 

653 

769 

883 

1013 

2 

61 

149 

239 

347 

443 

563 

659 

773 

887 

1019 

3 

67 

151 

241 

349 

449 

569 

661 

787 

907 

1021 

5 

71 

157 

251 

353 

457 

571 

673 

797 

911 

1031 

7 

73 

163 

257 

359 

461 

577 

677 

809 

919 

1033 

11 

79 

167 

263 

367 

463 

587 

683 

811 

929 

1039 

13 

83 

173 

269 

373 

467 

593 

691 

821 

937 

1049 

17 

89 

179 

271 

379 

479 

599 

701  • 

823 

941 

1051 

19 

97 

181 

277 

383 

487 

601 

709 

827 

947 

1061 

23 

101 

191 

281 

389 

491 

607 

719 

829 

953 

1063 

29 

103 

193 

283 

397 

499 

613 

727 

839 

967 

1069 

31 

107 

197 

293 

401 

503 

617 

733 

853 

971 

1087 

37 

109 

199 

307 

409 

509 

619 

739 

857 

977 

1091 

41 

113 

211 

311 

419 

521 

631 

743 

859 

983 

1093 

43 

127 

223 

313 

421 

523 

641 

751 

863 

991 

1097 

47 

131 

227 

317 

431 

541 

643 

757 

877 

997 

1103 

53 

137 

229 

331 

433 

547 

647 

761 

881 

1009 

1109 

113.  A  Prime  Factor  of  a  number  is  a  prime  number  that  will 
exactly  divide  it ;  thus,  the  prime  factors  of  21  are  the  prime 
numbers  1,  3,  and  7. 

A  Composite  Factor  of  a  number  is  a  composite  number  (Art. 
41)  that  will  exactly  divide  it ;  thus,  the  composite  factors  of  24 
are  the  composite  numbers  4  and  6. 

Note  1 .  —  Unity  or  1  is  not  commonly  regarded  as  a  prime  factor,  since 
multiplying  or  dividing  any  number  by  1  does  not  alter  its  value  ;  it  will  be 
omitted  vfhen  speaking  of  the  prime  factors  of  numbers. 

Note  2.  —  No  direct  process  of  finding  prime  numbers  has  been  discovered. 
The  following,  fticts,  however,  will  aid  in  ascertaining  whether  a  number  is 
prime  or  not ;  and,  if  not  prime,  will  indicate  one  or  more  of  its  factors  : 

1.  2  is  the  only  even  prime  number. 

2.  2  is  a  fiictor  of  every  even  number. 

3.  3  is  a  factor  of  every  number  the  sum  of  whose  digits  3  will  exactly 
divide ;  thus,  15,  81,  and  546  have  each  3  as  a  factor. 

4.  4  is  a  fector  of  every  number  whose  two  right-hand  figures  4  will  exactly 
divide  ;  thus,  316,  532,  and  1724,  have  each  4  as  a  factor. 

5.  5  is  the  only  prime  number  having  5  for  a  unit  or  right-hand  figure. 

113.  What  is  a  prime  factor  ?  What  is  a  composite  factor  1  How  is  unity 
or  1  rpgarded?  Is  there  anv  direct  process  for  determining  prime  numbers? 
Which  is  the  only  even  prime  number?  Of  what  numbers  is  2  a  factor? 
Of  what  numbers  is  3  a  factor  1     Of  what  numbers  is  4  a  factor  ? 


132  PROPERTIES   OF  NUMBERS. 

6.  5  is  a  factor  of  every  number  whose  right-hand  figure  is  either  5  or  0; 
as,  15,  20,  &c. 

7.  C  is  a  factor  of  every  even  number  that  3  will  exactly  divide;  thus,  24, 
108,  and  360  have  each  6  as  a  factor. 

8.  7  is  a  factor  of  every  number  occupying  four  places  whose  two  right- 
hand  figures  are  contained  in  the  left-hand  figure  or  figures  exactly  3  times ; 
thus,  2107  and  3913  have  each  7  as  a  factor. 

9.  7  is  a  factor  of  every  number  occupying  three  or  four  places,  when  the 
two  right-hand  figures  contain  the  left-hand  figure  or  figures  exactly  5  times  ; 
thus,  840,  945,  and  1 155  have  each  7  as  a  factor. 

10.  8  is  a  factor  of  every  number  whose  three  right-hand  figures  8  will 
exactly  divide  ;  thus,  5072,  11240,  and  17128  have  each  8  as  a  factor. 

11.  9  is  a  factor  of  every  number  the  sum  of  whose  digits  9  will  exactly 
divide  ;  thus,  27,  432,  ifnd  20304  have  each  9  as  a  factor. 

12.  10  is  a  factor  of  every  number  whose  right-hand  figure  is  0  ;  as,  20, 
30,  &c. 

13.  7,  1 1,  and  13  are  factors  of  any  number  occupying  four  places  in  which 
two  like  figures  have  two  ciphers  between  them  ;  as,  3003,  4U04,  9Uu9,  &c. 

■    14.  Every  prime  number,  except  2  and  5,  has  1,3,  7,  or  9  for  the  right- 
hand  figure. 

114,   To  find  the  prime  factors  of  numbers. 

Ex.  1.  Find  the  prime  factors  of  24.  Ans.  2,  2,  2,  3. 


OPERATION.  "We  divide  by  2,  the  least  prime  number  great- 

"24  er  than   1,   and   obtain    the   quotient  12.     And 

T~T  since  12  is  a  composite  number,  we  divide  this 

^  ^  also  by  2,  and  obtain  a  quotient  6.     We  divide 

6  6  by   2,  and   obtain   3   for  a  quotient,  which  is 

a  prime  number.     The  several  divisors  and  the 

3  last  quotient,  all  being  prime,  constitute  all  the 
prime  factors  of  24,  which,  multiplied  together, 


equal  2X2X2X3  =  24. 

Rule.  —  Divide  the  given  number  by  any  prime  number,  greater  than 
1,  that  will  divide  it,  and  the  quotient,  if  a  composite  number,  in  the  same 
manner;  and  continue  dividing  until  a  prime  number  is  obtained  for  a 
quotient.  The  several  divisors  and  the  last  quotient  tvill  be  the  prime 
factors  required. 

Note.  —  The  composite  fivctors  of  any  number  may  be  found  by  multi- 
plying together  two  or  more  of  its  prime  factors. 

1 13.  Of  what  numbers  is  5  a  factor?  Of  what  is  6  a  factor  ?  Of  what  is 
7  a  factor  ?  Of  what  is  8  a  factor  T  Of  wiiat  is  9  a  factor  >  What  is  the 
riiilit-hand  figure  of  every  primes  number?  — 114.  'i'lic  rule  for  fiiuliiig  the 
prime  factors  of  numbers  J  IIow  may  the  composite  factors  of  numbers  bo 
found  1 


CANCELLATION.  133 

Examples  for  Practice. 

2.  What  are  the  prime  factors  of  36?  Ans.  2,  2,  3,  3. 

3.  What  are  the  prime  factors  of  48  ?       Ans.  2,  2,  2,  2,  3. 

4.  What  are  the  prime  factors  of  5G  ?  Ans.  2,  2,  2,  7. 

5.  What  are  the  prime  factors  of  144?    Ans.  2,  2,  2,  2,  3,  3. 

6.  Find  the  prime  factors  of  3420  ?      Ans.  2,  2,  3,  3,  5, 19. 

7.  What  are  the  prime  factors  of  18500? 

Ans.  2,  2,  5,  5,  5,  37. 

8.  What  are  the  prime  factors  of  199  Go  ? 

Ans.  3,  5,  11,  11,  11. 

9.  What  are  the  prime  factors  of  12496  ? 

Ans.  2,  2,  2,  2,  11,  71. 

10.  What  are  the  prime  factors  of  17199  ? 

Ans.  3,  3,  3,  7,  7,  13. 

11.  What  are  the  prime  factors  of  7800  ? 

Ans.  2,  2,  2,  3,  5,  5,  13. 

CANCELLATION. 

115.  If  the  dividend  and  divisor  are  both  divided  by  the  same 
number,  the  quotient  is  not  changed.  Thus,  if  the  dividend  is 
20  and  the  divisor  4,  the  quotient  will  be  5.  Now,  if  we  divide 
the  dividend  and  divisor  by  some  number,  as  2,  we  obtain  10 
and  2  respectively ;  and  10  -i-  2  =  5,  the  same  as  the  original 
quotient. 

Also,  if  the  dividend  and  divisor  are  both  multiplied  by  the  same 
number,  the  quotient  is  not  changed. 

116.  If  ci  factor  in  any  number  is  canceled,  the  number  is 
divided  by  that  factor.  Thus,  if  15  is  the  dividend  and  5  the 
divisor,  the  quotient  will  be  3.  Now,  since  the  divisor  and  quo- 
tient are  the  two  factors,  which,  being  multiplied  together,  pro- 
duce the  dividend  (Art.  50),  if  we  cross  out  or  cancel  the  foctor 
5,  the  remaining  3  is  the  quotient,  and  by  the  operation  the 
dividend  15  has  been  divided  by  5. 

117.  Cancellation  is  the  method  of  shortening  arithmetical 
operations  by  rejecting  any  factor  or  factors  common  to  the  divisor 
and  dividend. 


115.  What  is  the  effect  on  the  quotient  when  the  dividend  and  divisor  are 
divided  by  the  same  number "?  — 116.  The  etFect  of  canceling  a  factor  of 
any  number  1  —  117.  What  is  cancellation  'i 
12 


134  PROPERTIES  OF  NUMBERS. 

Ex.  1.  A  man  sold  25  hundred  weight  of  iron  at  5  dollars  per 
hundred  weight,  and  expended  the  money  for  flour  at  5  dollars 
per  barrel ;  how  many  barrels  did  he  purchase  ? 

Ans.  25  barrels. 

OPERATION.  "VVe  first  indicate  by  their  signs  the 

Dividend   ^  X  25  ^        multiplication  and  division  required  by 

=  2  5.       the  question.     Then,  observing  5  to  be 

Divisor  $  a  common    factor   of  the   divisor   and 

dividend,  we  divide  the  divisor  and 
dividend  by  this  factor,  or,  which  is  the  same  thing,  cancel  or  reject  it 
in  both,  and  obtain  25  lor  the  quotient. 

2.  Divide  the  product  of  12,  7,  and  5  by  the  product  of  5,  4, 
and  2.  Ans.  10^. 

OPERATION. 

Dividend  ^^XTX  5  ^21^ 
Divisor       ^X4X2  2  ^^ 

Finding  4  in  the  divisor  to  be  a  factor  of  1 2  in  the  dividend,  we 
divide  12  by  4,  canceling  these  numbers,  and  use  the  3  instead  of  12. 
The  factor  5,  common  to  both  dividend  and  divisor,  having  been  can- 
celed, we  divide  the  product  of  the  remaining  factors  in  the  dividend 
by  the  product  of  those  in  the  divisor,  and  obtain  the  quotient  10^. 

3.  Divide  the  product  of  8,  5]  16,  and  21  by  the  product  of 
10,  4,  12,  and  7. 

OPERATION. 
4 

Dividend  $X$Xt^X^t       ,^,.     , 

=  4,  Quotient. 

Divisor    X0X4.XX^Xli 

Tlie  product  of  the  factors  8  and  5  in  the  dividend  is  equal  to  the 
product  of  10  and  4  in  the  divisor ;  therefore  we  cancel  these  factors. 
Finding  16  in* the  dividend  and  12  in  the  divisor  may  be  exactly 
divided  by  4,  they  are  canceled,  and  use  made  of  the  quotients. 
Again,  as  the  product  of  the  factors  3  and  7  of  tlie  divisor  equals  the 
21  of  the  dividend,  we  cancel  the  3,  7,  and  21.  The  factor  4  alone 
remaining  is  the  quotient. 

117.  How  do  vou  arrftngc  tlio  diviilond  and  divisor  for  cancoUfition  ?  ITow 
do  you  then  proceed  f  Is  the  fnctor  5,  in  Ex.  1,  reduced  to  0  or  1  hy  heing 
canceled  ?  How  do  you  proceed  wlien  a  numl)er  in  tlie  dividend  and  anotlier 
in  the  divisor  have  a  coiumon  factor  ?  How  do  you  proceed  when  the  pro- 
ducts of  two  or  more  factors  in  the  dividend  and  divisor  arc  alike  1 


CANCELLATION.  135 

Rule.  —  Cancel  the  factor  or  factors  common  to  the  dividend  and 
divisor,  and  then  divide  the  product  of  the  factors  remaining  in  the  divi- 
dend by  the  product  of  tnose  remaining  in  the  divisor. 

Note  I.  —  In  arranging  the  numbers  for  cancellation,  the  dividend  may 
be  written  above  tlie  divisor  with  a  horizontal  line  between  them,  as  in  divis- 
ion (Art.  47)  ;  or,  as  some  prefer,  the  dividend  may  be  written  on  the  right 
of  the  divisor,  with  a  vertical  line  between  them. 

Note  2. —  Canceling  a  factor  does  not  leave  0,  but  the  quotient  1,  to  take 
its  i)lace,  since  rejecting  a  factor  is  tlie  same  as  dividing  by  that  factor  (Art. 
116).     Therefore,  for  every  factor  canceled  1  remains. 

Examples  for  Practice. 

4.  Divide  42  X  19  by  19.  Ans.  42. 

5.  Divide  the  product  of  8,  6,  and  3,  by  the  product  of  6,  3, 
and  4.  Ans.  2. 

6.  Divide  the  product  of  17,  6,  and  2,  by  the  product  of  6,  2, 
and  17.  Ans.  1. 

7.  Sold  15  pieces  of  shirting,  and  in  each  piece  there  were  30 
yards,  for  which  I  received  10  cents  per  yard ;  expended  the 
money  for  10  pieces  of  cahco,  eacli  containing  15  yards  ;  what 
was  the  calico  per  yard  ?  Ans.  30  cents. 

8.  Divide  the  product  of  12,  7,  and  5,  by  the  product  of  2,  4, 
and  3.  Ans.  17^. 

9.  Divide  the  product  of  20,  13,  and  9,  by  the  product  of  13, 
16,  and  1.  Ans.  11^. 

10.  Divide  the  product  of  9,  8,  2,  and  14,  by  the  product  of 
3,  4,  6,  and  7.  Ans.  4. 

11.  Divide  the  product  of  16,  5,  10,  and  18,  by  the  product  of 
8,  6,  2,  and  12.  Ans.  12^. 

12.  Divide  the  product  of  22,  9,  12,  and  5,  by  the  product 
of  3,  11,  G,  and  4.  Ans.  15. 

13.  Divide  the  product  of  25,  7,  14,  and  36,  by  the  product 
of  4,  10,  21,  and  54.  Ans.  l-i^. 

14.  Divide  the  product  of  26,  72,  81,  and  12,  by  the  product 
of  36,  13,  24,  and  54.  Ans.  3. 

15.  Divide  the  product  of  8,  5,  3,  16,  and  28,  by  the  product 
of  10,  4,  12,  4,  and  7.  Ans.  4. 

10.  Divide  the  product  of  8,  4,  9,  2,  12,  16,  and  5,  by  the 
product  of  4,  6,  6,  3,  8,  4,  and  20.  Ans.  2. 

17.  Divide  the  product  of  6,  15,  16,  24,  12,  21,  and  27,  by 
the  product  of  2,  10,  9,  8,  36,  7,  and  81.  Ans.  8. 

117.  The  rule  for  cancellation  ?  How  may  the  numbers  be  arranged  for 
canceling"?     What  remains  for  every  factor  canceled? 


136  PROPERTIES   OF   l^UMBERS. 

A   COMMON  DIVISOR. 

118.  A  Common  Divisor  of  two  or  more  numbers  is  any 
number  that  will  divide  them  without  a  remainder ;  thus,  2  is 
a  common  divisor  of  2,  4,  6,  and  8. 

119.  To  find  a  common  divisor. 

Ex.  1.   What  is  the  common  divisor  of  10,  15,  and  25  ? 

Ans.  5. 

OPERATION.  We  resolve  each  of  the  given  numbers  into  two 

10  =  5   X  2  factors,  one   of  which  is  common   to   all   of  them. 

2  5  __  5   y  3  In  the  operation  5  is  the  common  factor,  and  there- 

f)  c  TV  5  ^^^*^  must  be  a  common  divisor  of  the  numbers. 

Rule. — Resolve  each  of  the  given  numbers  into  two  factors,  one  of 
which  is  common  to  all  of  them,  and  this  common  factor  is  a  common 
divisor. 

Examples  for  Practice. 

2.  Wliat  is  the  common  divisor  of  3,  9,  18,  24  ?         Ans.  3. 

3.  What  is  the  common  divisor  of  4,  12,  16,  28  ? 

Ans.  2  or  4. 

120.  A  divisor  of  any  factor  of  a  number  is  a  divisor  of  the 
number  itself.  Thus  3,  a  divisor  of  9,  a  factor  of  45,  is  a  divisor 
of  45  itself. 

121.  A  common  divisor  of  two  numbers  is  a  divisor  of  their 
sum  and  of  their  difference.  Thus  4,  a  common  divisor  of  16 
and  12,  is  a  divisor  of  their  sum,  28,  and  of  their  difference,  4. 

122.  A  common  divisor  of  the  remainder  and  the  divisor  is 
a 'divisor  of  the  dividend.  Thus,  in  a  division  having  12  for 
remainder,  36  for  divisor,  and  48  for  dividend,  12,  a  common 
divisor  of  the  12  and  the  36,  is  also  a  divisor  of  tlie  48. 

TUE   GREATEST   COMMON  DIVISOR. 

123.  The  Greatest  Common  Divisor  of  two  or  more  numbers  is  the 

greatest  number  that  will  divide  each  of  them  without  a  remain- 
der.    Thus  6  is  the  greatest  common  divisor  of  12,  18,  and  24. 


118.  Wliat  is  a  roinition  divisor  of  two  or  more  numhors? — 119.  The 
rule?  —  121.  Of  wliiit  is  the  c-ommon  divisor  of  two  mimlicrs  a  divisor'?— - 
122.  Of  what  is  a  common  divisor  of  the  less  of  two  miniliors  and  of  their 
difference  a  divisor  ?  —  123.  What  is  the  grcutcst  common  divisor  of  two  or 
more  aumbcra  1 


GREATEST   COMMON   MEASURE,  187 

124.    To  find  the  greatest  common  divisor. 

Ex.  1.    What  is  the  greatest  common  divisor  or  measure  of 
84  and  132  ?  Ans.  12. 

FIRST  OPERATION.  Rcsolvlnji  the   numbers  into   their 
84  =  2X2X3X      7  prime  factors  (Art.  114),  thus,  84  = 
13  2  =  2X2X3X11  2  X  2   X   3   X    7,  and  132  =  2  X  2 
^            f^  X  3  X  11,  we  find  the  factors  2X2 
2  X.  2  y(_  o  =  1  2.  v,^3  j^j.g  common  to  both.     Since  only 
these  common  factors,  or  the  product  of  two  or  more  of  such  fac- 
tors, will  exactly  divide   both  numbers,  it  follows  that  the  product  of 
all  their  common  prime  factors  must  he  the  greatest  factor  that  ivill  ex- 
actly divide  both  of  them.     Therefore  2X2X3  =  12  is  the  greatest 
common  divisor  required. 

The  same  result  may  be  obtained  by  a  sort  of  trial  process,  as 
by  the  second  operation. 

SECOND  OPERATION.  Slucc    84   canuot  be    exactly 

84)  132  (1  divided    by   a    number    greater 

g  4  than  itself,  if  it  will  also  exactly 

—  divide  132,  it  will  be  the  greatest 

4  8)84(1               .  common  divisor  sought.     But,  on 

4  8  trial,  we  find  84  will  not  exactly 

rr".    ^  Q  /  1  divide  132,  there  being  a  remain- 

d  b  )  4  8  (  1  jer,  48.     Therefore  84  is  not  a 

3  6  common  divisor  of  the  two  num- 

12)36(3     ^^i:',;    ,  ,.  . 

■^  n  n  »Ve   know  a   common   divisor 


36 


of  48  and  84  will  also  be  a  divisor 


of  132  (Art.  122).  We  next  try  to  find  that  divisor.  It  cannot  be 
greater  than  48.  But  48  Avill  not  exactly  divide  84,  there  being  a 
remainder,  36 ;  therefore  48  is  not  the  greatest  common  divisor. 

Again,  as  the  common  divisor  of  36  and  48  will  also  be  a  divisor  of 
84  (Art.  122),  we  try  to  find  that  divisor,  knowing  that  it  cannot  be 
greater  than  36.  But  36  will  not  exactly  divide  48,  there  being  a  re- 
mainder, 12  ;  therefore  36  is  not  the  greatest  common  divisor. 

As  before,  the  common  divisor  of  12  and  36  will  be  a  divisor  of  48 
(Art.  122);  we  make  a  trial  to  find  that  divisor,  knowing  that  it  can- 
not be  greater  than  12,  and  find  12  will  exactly  divide  36.  Therefore 
12  is  the  greatest  common  divisor  required. 

Rule  1. — Resolve  the  given  numbers  into  their  prime  factors.  The 
product  of  all  the  factors  common  to  the  several  numbers  tvill  be  the 
greatest  common  divisor.     Or, 

Rule  2.  —  Divide  the  greater  number  b>j  the  less,  and  if  there  be  a 

1 24.  What  are  the  rules  for  finding  the  greatest  common  divisor  of  two  or 
more  numbers  1 

12* 


138  PROPERTIES   OF  NUMBERS. 

remainder  divide  the  preceding  divisor  by  it,  and  so  continue  dividing 
until  nothing  remains.  The  last  divisor  will  be  the  greatest  common 
divisor. 

Note.  —  When  the  greatest  common  divisor  is  required  of  more  than  two 
numbers,  find  it  of  two  of  them,  and  then  of  that  common  divisor  and  of 
one  of  the  other  numbers,  and  so  on  for  all  the  given  numbers. 

Another  method  is  to  divide  the  numbers  by  any  factor  common  to  them  all; 
and  so  continue  to  divide  till  there  are  no  longer  any  common  factors ;  and  the 
product  of  all  the  common  factors  will  be  the  greatest  common  divisor   required. 

Examples  for  Practice. 

2.  What  is  the  greatest  common  divisor  of  85  and  95  ? 

Ans.  5. 

3.  What  is  the  greatest  common  divisor  of  72  and  1 68  ? 

Ans.  24. 

4.  What  is  the  greatest  common  divisor  of  119  and  121  ? 

Ans.  1. 

5.  What  is  the  greatest  common  divisor  of  12,  18,  24,  and 
30?  Ans.  6. 

6.  Having  three  rooms,  the  first  12  feet  wide,  the  second  15 
feet,  and  the  third  18  feet,  I  wish  to  purcliase  a  roll  of  the  widest 
carpeting  that  will  exactly  fit  each  room  without  any  cutting  as 
to  width.     How  wide  must  it  be  ?  Ans.  3  feet. 

A  COMMON  MULTIPLE. 

125.  A  Multiple  of  a  number  is  a  number  that  can  be  divided 
by  it  without  a  remainder ;  thus  6  is  a  multiple  of  3. 

126.  A  Common  Multiple  of  two  or  more  numbers  is  a  num- 
ber that  can  be  divided  by  each  of  them  without  a  remainder; 
thus  12  is  a  common  multiple  of  3  and  4. 

127.  The  least  Common  Multiple  of  two  or  more  numbers  is 
the  least  number  that  can  be  divided  by  eacli  of  them  without  a 
remainder  ;  thus  30  is  the  least  common  multiple  of  10  and  15. 

Note.  —  A  multiple  of  a  number  contains  all  the  prime  factors  of  that 
number ;  and  the  common  multiple  of  two  or  more  numbers  contains  all  the 
prime  factors  of  each  of  the  numbers.  Therefore,  the  least  common  multiple 
of  two  or  more  numbers  must  be  the  least  number  that  will  contain  all  the 
prime  factors  of  them,  and  none  others.  Hence  it  will  have  cacli  i>rimc  factor 
taken  only  the  greatest  number  of  times  it  is  found  in  any  of  the  several 
numbers. 

125.  What  is  a  mul»ii)le  of  a  number  ?  —  127.  The  least  common  multiple 
of  two  or  more  numbers  1 


LEAST   COMMON   MULTIPLE.  139 

128.   To  find  the  least  common  multiple. 

Ex.  1.   What  is  the  least  common  multiple  of  6,  9,  12  ? 

Ans.  36. 

FIRST  OPERATION.  Kesolving  the  numbers  into  their  prime 

6  =  2X3  factors,  —  thus,  6  =  2  X  3,  and  9  =  3  X 

9=3X3  3,  and  12  =  2  X  2  X  3,  —  we  find  their 

j^2  =  2X2X3       different  prime  factors  to  be  2  and  3.     The 

9v9v^v^ Sr     g''^^*''^^''  number  of  times  the  2  occurs  as  a 

^X^X«>X^  «50  factor  in  any  of  the  numbers  is  twice,  as 
2  X  2  in  12;  and  the  greatest  number  of  times  the  3  occurs  in  any 
of  the  numbers  is  also  twice,  as  3  X  3  in  9.  Hence  2X2X3X3 
must  be  all  the  prime  factors  that  are  necessary  in  composing  6,  9,  and 
12  ;  and,  consequently,  the  product  of  these  factors  must  be  the  least 
number  that  can  be  exactly  divided  by  C,  9,  and  12.  Therefore 
2X2X3X3  =  36  is  the  least  conuiion  multiple  required. 

Having:   arranged  the  numbers  on  a 
horizontal  line,  we  divide  by  3,  a  prime 
number  that  will  divide  all  of  them  with- 
out a  remainder,  and  write  the  quotients 
in  a  line  below.     We  next  divide  by  2, 
a  prime  number,  writing  down  the  quo- 
tients and  undivided  numbers  as  belbre. 
Then,  since  these  numbers  are  prime  to 
each  other,  we  multiply  together  the  divisors  and  the  numbers  on  the 
lower  line,  which  are  all  the  prime  factors  of  6,  9,  and  12,  and  thus 
obtain  36  for  the  least  common  multiple. 

KuLE  1.  —  Rexolve  the  given  numbers  into  their  prime  factors.  The 
product  of  these  factors,  talcing  each  factor  the  greatest  number  of  times 
it  occurs  in  any  of  the  numbers,  will  be  the  least  common  multiple.     Or, 

Rule  2.  —  Having  arranged  the  numbers  on  a  horizontal  line,  divide 
by  such  a  prime  number  as  will  divide  most  of  them  without  a  remainaer, 
and  icrite  the  quotients  and  undivided  numbers  in  a  line  beneath.  So 
continue  to  divide  until  no  prime  number  greater  than  1  will  divide  two  or 
more  of  them.  The  product  of  the  divisors  and  the  numbers  of  the  line 
below  will  be  the  least  common  multiple.  , 

Note  1 .  —  When  numbers  are  prime  to  each  other,  their  product  is  their 
least  common  multiple. 

Note  2.  —  When  any  of  the  given  numbers  is  a  factor  of  any  of  the  others 
it  may  be  canceled. 


SECOND   OPERATION. 

3 

6     9    12 

2 

2     3       4 

1     3       2 

X  2  > 

(  3  X  2  =  36 

128.   What  are  the  rules  for  finding  the  least  common  multiple  ? 


140  FRACTIONS. 

ExAJiiPLEs  FOR  Practice. 

2.  Wliat  is  the  least  common  multiple  of  7,  14,  21,  and  15? 

Ans.  210. 

OPERATION. 

rf    -i  A    o]     15       Since   7  is  a  factor  of  14,  another  of  the 

^_ numbers,  we  cancel  it ;  and  since  3  is  a  factor 

2       2|    1  5  of  15,  we  also  cancel  that  (Note  2)  ;  thus  the 
'  work  is  rendered  shorter. 

7X  2  X  15  =  210 

3.  What  is  the  least  common  multiple  of  3,  4,  5,  6,  7,  and  8  ? 

Ans.  840. 

4.  What  is  the  least  number  that  10,  12,  16,  20,  and  24  will 
divide  without  a  remainder  ?  Ans.  240. 

5.  What  is  the  least  common  multiple  of  9,  8,  12,  18,  24,  36, 
and  72  ?  Ans.  72. 

6.  Five  men  star!  from  the  same  place  to  go  round  a  certain 
island.  The  first  can  go  round  it  in  10  days  ;  the  second,  in  12 
days;  the  tliird,  in  16  days  ;  the  fourth,  in  18  days  ;  the  fifth  in 
20  days.  In  what  time  will  they  all  meet  at  the  place  from 
which  they  started?  Ans.  720  days. 


FRACTIONS. 


129.    A  Fraction  is  an  expression  denoting  one  or  more  equal 
parts  of  a  unit. 

Fraction  is  derived  from  the  Latin  frango,  to  hreaJc. 

Fractions  are  of  two  kinds.  Common  and  Decimal. 

COMMON  FRACTIONS. 

1.30.    A  Common   Fraction  is   expressed  by  two  numbers,  one 
written  over  the  other,  with  a  line  between  them. 

The  number  below  the  line  is  called  the  denominator ;  and  the 
number  above,  the  numerator. 

f  Numerator,     3,  Three 

Thus,    <.  — 

(  Denominator,  5,  Fifths. 

129.  What  is  n  fraction  ?  From  \vh:U  is  tlie  term  derived,  and  what  does 
it  sif^nifv?  IIow  miiiiy  i<inds  of  fractions,  and  what  arc  they  calli'd  ?  — 
1.30.  How  is  a  common  fiaction  expressed  ?  What  is  the  number  below  the 
lino  called  ?     The  number  above  the  lino  ? 


COMMON   FRACTIONS.  •        141 

The  Denominator  shows  into  how  many  parts  the  whole  number 
is  divided,  and  gives  a  name  to  the  fraction. 

The  Numerator  shows  how  many  of  these  parts  are  taken,  or 
expressed  by  the  fraction. 

A  Proper  Fraction  is  one  whose  numerator  is  less  than  the  de- 
nominator ;  as,  ^. 

An  Improper  Fraction  is  one  whose  numerator  is  equal  to,  or 

greater  than,  the  denominator ;  as,  f,  f . 

A  Mixed  Number  is  a  whole  number  with  a  fraction;  as,  7y\,  5|. 

A  Simple  or  Single  Fraction  has  but  one  numerator  and  one  de- 
nominator, and  may  be  either  proper  or  improper;  as,  |,  ■§. 

A  Compound  Fraction  is  a  fraction  of  a  fraction,  connected  by  the 
word  of;  as,  ^  of  |^  of  |. 

A  Complex  Fraction  is  a  fraction  having  a  fraction  or  a 
mixed  number  for  its  numerator  or  denominator,  or  both ;  as, 

1'91'11'9-rV 

131.  The  Terms  of  a  fraction  are  its  numerator  and  denom« 
inator. 

The  Unit  of  a  Fraction  is  the  unit  or  whole  thing  divided. 

A  Fractional  Unit  is  one  of  the  equal  parts  into  which  the  unit 
of  the  fraction  is  divided. 

A  whole  number  may  be  expressed  fractionally,  by  writing  1 
for  the  denominator.  Thus,  5  may  be  written  ^,  and  read  5 
ones;  and  9  may  be  written  |,  and  read  9  ones. 

132.  Fractions  originate  from  division ;  the  numerator  an- 
swers  to  the  dividend,  and  the  denominator  to  the  divisor.  Thus, 
when  we  divided  479956  by  6  (Art.  49,  Ex.  12),  we  had  a 
remainder  of  4,  which  could  not  be  divided  by  6,  and  therefore 
we  wrote  it  over  the  divisor,  with  a  line  between  them.  This 
expression  originating  from  division  is  a  fraction ;  the  number 
above  the  line  being  the  numerator,  and  the  one  below  the  de- 
nominator. 

130.  What  does  the  denominator  of  a  fraction  show  ?  "What  does  the 
numerator  show  ?  What  is  a  proper  fraction  ?  An  improper  fraction  ?  A 
mixed  number  ?  A  simple  fraction  ?  A  compound  fraction  ?  A  complex 
fraction  '^  — 131 .  What  are  the  terms  of  a  fraction  ?  What  is  the  unit  of  a 
fraction  ?  How  may  a  whole  number  be  expressed  fractionally  ?  From 
what  do  fractions  originate  ? 


142  COMMON   FRACTIONS. 

133.  Tlie  value  of  a  fraction  is  the  quotient  arising  from  the 
division  of  the  numerator  by  the  denominator.  Thus,  the  value  of 
f ,  or  6  -7-  2,  is  3  ;  and  the  value  of  f ,  or  3  -t-  4,  is  f .  • 

REDUCTION. 

134.  Reduclion  of  Fractions  is  the  process  of  changing  their 
form  without  altering  tlieir  value. 

A  fraction  is  in  its  lowest  terms,  when  its  terms  are  prime  to 
each  other.     (Art.  112.) 

135i   To  reduce  a  fraction  to  its  lowest  terms. 

Ex.  1.  Reduce  ^^  to  its  lowest  terms.  Ans.  ^. 

OPERATION.  We  divide  the  terms  of  the  fraction  by  2,  a  factor 

2  )  -6=   3     common  to  them  both,  and  obtain  |^.      We  divide, 

ox      3  1     again,   both  terms  of  4  by  3,  a  factor  common  to 

/  ,  ^  ^  them,  and  obtain  \.  Isow,  as  1  and  3  are  numbers 
prime  to  each  other,  the  fraction  \  is  in  its  lowest  terms.  The  same 
result  would  have  been  produced,  if  we  had  divided  the  terms  by  6, 
the  gi'eatest  common  divisor. 

Since  the  numerator  and  denominator  of  a  fraction  correspond  to 
the  dividend  and  divisor  in  division  (Art.  132),  dividing  both  by  the 
same  number,  or  canceling  equal  factors  in  both  (Ai't,  115),  changes 
only  the  form  of  the  fraction,  while  the  value  expressed  remains  the 
same.     Therefore, 

Dividing  the  numerator  and  denominator  of  a  fraction  hy  the  same 
number  does  not  alter  the  value  of  the  fraction. 

Rule.  —  Divide  the  numerator  and  denominator  hy  any  number 
greater  than  1,  that  will  divide  them  both  without  a  remainder,  and  thus 
proceed  until  they  are  prime  to  each  other.     Or, 

Divide  both  terms  of  the  fraction  by  their  greatest  common  divisor. 

Examples  for  Practice. 

2.  Reduce  ^  to  its  lowest  terms.  Ans.  \. 

3.  Reduce  -t^  to  its  lo^vest  terms.  Ans.  f . 

4.  Reduce  f  f  to  its  lowest  terms.  Ans.  \. 

5.  Reduce  yW  to  its  lowest  terms.  Ans.  f . 

6.  Reduce  ^xt  to  its  lowest  terms.  Ans.  ^. 

7.  Reduce  ^gf  to  its  lowest  terras.  Ans.  Jff. 

8.  Reduce  -f^j  to  its  lowest  terms.  Ans.  \. 

133.  What  is  the  v.ihic  of  a  fraption  ?  —  134.  Wlint  is  reduction  of  frac- 
tions ?  ■  "NVhon  is  ft  friiction  in  its  lowest  tenns  ?  —  l.'J.'i.  Wiiy  does  diviiliiip; 
both  terms  of  a  fraction  hy  tlic  same  number  not  ahcr  tho  value  ?  Has  i 
the  same  value  as  ^^  ?     Why  ?     Repeat  the  rule. 


I 


REDUCTION.  143 

9.  Reduce  i^ffi  to  its  lowest  terms.  Ans.  |f-f |-. 

10.  What  is  the  lowest  expression  of  fyf  ^  Ans.  ^J|. 

136.    To  reduce  a  mixed  number  to  an  improper  frac- 
tion. 

Ex.  1.  In  7|  how  many  fifths  ?  Ans.  ^-^. 

OPERATION. 

7  -I  Since  there  are  5  fifths  in  1  whole  one,  there  will 

5  be  5  times  as  many  Jifths  as  whole  ones ;  therefore, 

— -  in  7  there  are  35  fit\hs,  and  the  3  fifths  being  added 

o  o  nitus.  make  38  fifths,  which  are  expressed  thus,  ^. 


3  8  fifths  =  -3f . 

Rule.  —  Multiply  the  whole  number  hy  the  denominator  of  the  fraction^ 
and  to  the  product  add  the  numerator,  and  place  the  sum  over  the  (jiven 
denominator. 

Note.  —  To  rerlnce  a  whole  number  to  a  fraction  of  the  same  vahie, 
having  a  given  denominator,  we  multiply  the  whole  number  by  the  yiven 
denominator,  and  make  the  product  the  numerator ;  thus  5,  reduced  to  a  frac- 
tion, having  3  for  a  denominator,  becomes  ^^-. 

Examples  for  Practice. 

< 

2.  In  8f  dollars  how  many  sevenths  ?  Ans.  -^. 

3.  In  3|-  oranges  how  many  fourths  ?  Ans.  ^^-. 

4.  In  9y*^  gallons  how  many  elevenths?  Ans.  -^f-^-. 

5.  Reduce  8y\  to  an  improper  fraction.  Ans.  ^\. 

6.  Reduce  lo/j  to  an  improper  fraction.  Ans.  JyV-- 

7.  In  18|-  how  many  ninths  ?  Ans.  ^^K 

8.  In  leiyVi^  how  many  one  hundred  and  seventeenths? 

Ans.  -Lff f-l, 

9.  Change  43m  ^^  an  improper  fraction.  Ans.  -Vrr' 

10.  AVhat  improper  fraction  will  express  21-^^  ?     Ans.  -\%°-. 

11.  Change  lHyJ-y  to  an  improper  fraction.       Ans.  -ifff-- 

12.  Change  125  to  an  improper  fraction.  Ans.  ^K 

13.  Change  25  to  an  improper  fraction,  having  6  for  a  de- 
nominator, j^ns.  ^1°: 

14.  Reduce  75  to  ninths.  Ans.  ^K 

15.  Change  343  to  the  form  of  a  fraction.  Ans.  ^^. 

16.  Reduce  84  to  fifteenths.  Ans.  ifl^. 


1  5 


_  136.  "What  is  the  rule  for  reducing  a  mixed  number  to  an  improper  frac- 
tion 1  The  reason  ?  How  do  you  reduce  a  whole  number  to  a  fraction  of 
the  same  value,  having  a  given  denominator  ? 


144  COMMON  FRACTIONS. 

137.    To  reduce  improper  fractions  to  "W'liole  or  mixed 
numbers. 

Ex.  ] .  How  many  dollars  in  ^^  dollars  ?  Ans.  $  2-j*g. 

OrERATION. 

1  6  ")  3  7  r  2-^  Since  IG  sixteenths  make  one  dollar,  there  will  be 

on  as  many  dollars  irt  37  sixteenths  of  a  dollar  as  37 


contains  times  16,  or  $  2j^^. 
5 

EuLE^  —  Divide  the  numerator  hy  the  denominator,  and  the  quotient 
tcill  be  the  whole  or  mixed  number. 

Examples  for  Practice. 

2.  Reduce    -^/-    to  a  whole  number.  Ans.  12. 

3.  Change  J/^^-  to  a  mixed  number.  Ans.  lO^V 

4.  Change  -VyV-  to  a  mixed  number.  Ans.  lOy-f^. 

5.  Change  VrV'^^  ^  mixed  number.  Ans.  Ifff. 

6.  Reduce  -Liyo-O-  to  a  mixed  number.  Ans.  142f. 
,      7.  Reduce  §f  |  to  a  whole  number.  Ans.  1. 

8.  Change  ^^^  to  a  whole  number.  Ans.  567. 

9.  Reduce  ^^^-   to  a  mixed  number.  Ans.  9^f . 
10.  Reduce  Vs*^  ^^  ^  mixed  number.                    Ans.  4^1^^. 

138.  To  reduce  a  compo\ind  fraction  to  a  simple  frac- 
tion. 

Ex.  1.    Reduce  f  of  -^^  to  a  simple  fraction.  Ans.  ||. 

OPERATION.  If  ^1^  be  divided  into  5  equal  parts,  one  of 

i  X  TT  =^  ft     these  parts  is  -^\ ;  and  if  -^  of  rr  be  3'^,  it  is 

evident  that  -^  of  -^j  will  be  7  times  as  much. 

7  times  -^^  is  -/-j  ;  and  if  \  of  /j-  be  -/-j,  |  of  y\  will  be  4  times 

as  much.     4  times  -/^  are  |f . 

Or,  by  multiplying  the  denominator  of  -j^j-  by  5,  the  denomi- 
nator of  ^,  it  is  evident  we  obtain  ^  of  -/j  =  -^-i;,  since  the  parts 
into  which  the  number  or  thing  is  divided  are  5  times  as  many, 
and   consequently  only   ^  as  large  as  before.     Again,  since  -^ 

137.  What  is  the  rnl(>  for  rcihicinsr  improper  frnetions  to  whole  or  mixed 
numbers?  A  reason  for  tlic  rule.  —  138.  How  do  you  reduce  a  compound 
fraction  to  a  simple  one  1     Tiie  reason  for  the  operation  1 


REDUCTION.  145 

of  T^-j-  =  /^,  ^  of  /y  will  be  4  times  as  much ;  and  4  times 
/&  =  ft-  This  procei5S  will  be  seen  to  be  precisely  like  the 
operation. 

Ex.  2.    Reduce  f  of  f  of  f  of  f  of  ^y  to  a  simple  fraction. 

Ans.  ■^. 

OPERATION  BY    CANCELLATION.      . 

o 

The  numerators  and  denomina- 

3X4XpX0X      7 2       tors  wliich  are  common  factors  we 

As^  A  \/~<i^ ih\/  1  1  "'  i~i  cancel  according  to  the  principles 
^X0X;?XyXli        ll     of  cancellation.     (Art.  117.) 

Rule.  —  Multiply  all  the  numerators  together  for  a  new  numerator, 
and  all  the  denominators  for  a  new  deiiominator. 

Note  1.  —  All  whole  and  mixed  numbers  in  the  compound  fraction  must 
be  reduced  to  improper  fractions,  before  muhipljing  the  numerators  and 
denominators. 

Note  2.  —  "When  there  are  factors  common  to  both  numerator  and  denom- 
inator, they  may  be  canceled  in  the  operation. 

ExAMiPLES    FOR    PRACTICE. 

3.  What  is  f  of  I  of  f  ?  Ans.  yVr  =  ^f- 

4.  What  is  I  of  ^\  of  7  ?  Ans.  5^V' 

5.  What  is  f  of  y^y  of  f  of  f  ?  Ans.  -f^j.. 

6.  Change  \^  of  |-  of  f  of  ^V  of  7  to  a  simple  fraction. 

Ans.  sYsV* 

7.  Required  the  value  of  f  of  ^  of  4t  of  s-^-  ^f  5f . 

Ans.  f . 

8.  Reduce  -i  of  f  of  ^^  of  |  of  ^  to  a  simple  fraction. 

Ans.  Y^. 

9.  Reduce  f  of  -^-^  of  ^  of  -fjy  of  4^  to  a  simple  fraction. 

Ans.  If. 

10.  Reduce  -j-f  of  f  of  -/y  to  a  simple  fraction.         Ans.  ^^. 

11.  Reduce  -^  of  ff  of  ^f  of  9  f  to  a  whole  number. 

Ans.  3. 

12.  Reduce  ^  of  y^  of  -^  of  8|^  of  -^  to  a  simple  fraction. 

Ans.  -f^. 

138.  When  there  are  common  factors  in  the  numerator  and  denominator, 
how  may  the  operation  be  shortened  ?     The  rule  ?    What  must  be  done  with 
all  whole  and  mixed  numbers  in  the  compound  fraction?     How  may  the 
operation  be  shortened  by  canceling  ? 
13 


u 

u 

1  =  llg 

(( 

(( 

i-=m 

146  COMMON   FRACTIONS. 

A  COMMON  DENOMINATOPw 

139.  A  Common  Denominator  of  two  or  more  fractions  is  a 

common  multiple  of  their  denominators. 

The  Least  Common  Denoraiiiator  of  two  or  more  fractions  is  the 
least  common  multiple  of  their  denominators. 

Note.  —  Fractions  haA'e  a  common  denominator,  when  all  their  denomi- 
nators are  alike. 

110.    To  reduce  fractions  to  a  common  denominator. 
Ex.  1.    Reduce  f ,  f,  mad  ^  to  a  common  denominator. 

An^      14  4      160      168 
OPERATION. 

3X6X8=14  4,  new  numerator,    f  =  -f  f  #. 

5X4X8=16  0, 

7X4X6^168, 

4X6X8  =  19  2,  common  denominator. 

"We  first  multiply  the  numerator  of  f  by  the  denominators  6  and  8, 
and  obtain  144  for  a  new  numerator.  We  next  multiply  the  numer- 
ator of  I  by  the  denominators  4  and  8,  and  obtain  IGO  for  a  new 
numerator ;  and  then  we  multiply  the  numerator  of  ^  by  the  denom- 
inators 4  and  6,  and  obtain  ICS  for  a  new  numerator.  Finally,  we 
multiply  all  the  denominators  together  for  a  common  denominator,  and 
write  it  under  the  several  numerators,  as  in  the  operation. 

By  this  process,  since  the  numerator  and  denominator  of  each  frac- 
tion are  multiplied  by  the  same  numbers,  only  the  form  of  the  fraction 
is  changed,  while  the  quotient  arising  from  dividing  the  numerator  by 
the  denominator,  or  the  value  of  the  fraction  (Ai-t.  1.33),  remains  the 
same.     Therefore, 

Multiplying  the  numerator  and  denominator  of  a  fraction  hy[  the  same 
number  does  not  alter  the  value  of  the  fraction. 

Rule.  —  Multiply  each  numerator  by  all  (he  denominators  except  its 
own,  for  the  new  numerators;  and  all  the  denominators  together  for  a 
common  denominator. 

Note  1.  —  Compound  fractions,  if  any,  must  first  be  reduced  to  simple 
ones,  and  whole  or  mixed  numbers  to  improper  fi-actions. 

Note  2.  —  Fractions  may  often  bo  reduced  to  lower  terms,  without  de- 
stroying their  common  denominator,  by  dividing  all  their  numerators  and 
deiioniinators  l)y  a  common  divisor. 

139.  What  is  a  common  denominator  of  two  or  more  fractions?  What  is 
the  least  common  denominator  ?  Wlien  have  fractions  a  common  denomi- 
nator ? —  140.  How  do  you  find  a  common  dniomiiiator  of  two  or  more 
fractions  1  Give  the  reason  of  the  operation.  Wliat  infi'rcuce  is  drawn  from 
it?  What  is  the  rule  for  fuiding  a  common  denominator?  How  may  frac- 
tions having  a  common  denominator  be  reduced  to  lower  terras  ? 


REDUCTION.  147 

Examples  for  Practice. 

2.  Reduce  f  and  f  to  common  denominators. 

Ant),  g-j*  24»  ^^  Ta'  T^* 

3.  Reduce  J,  f ,  and  ^  to  a  common  denominator. 

Ana    X"     72    45 

4.  Reduce  f ,  f ,  and  -^j  to  a  common  denominator. 

An«     352     231     280 
AUt..    gyg-,   ^yg^,  -g-j^. 

5.  Reduce  f ,  1%;  and  f  to  a  common  denominator. 

Ans     28  8      135      2J.6.     nv    32      15      24 
J\.ns.  S2i:  -S^iJ  ^2t»  Ol   Tjg,  Tj^,  -g^. 

6.  Reduce  ^,  f ,  f ,  and  f  to  a  common  denominator. 

Ans      160      384      840      2  40     or  -?-Q^    -A8       igS     -3  0 
-a-Ufc).  -g-gg-,  ^6(J'  FB^tf'  ?B^05  "^    120^?    12tJ)    IJTJJ    12XT* 

141.    To  reduce  fractions  to  their  least  common  denom- 
'nator. 

Ex.  1.  Reduce  f ,  f ,  and  -/g-  to  the  least  common  denominator. 

OPERATION. 

1  2,  least  common  denominator. 


0    0    12        3 

6 


4X2=    8,  new  numerator,    f 
2  X  5  =  10,    "  " 

1X7=7,"  " 


a 

fi 

;j 

12" 

5 

1  0 

■6 

12 

7 

7 

12 

1^ 

I 


12 
12,  least  common  multiple,  and  common  denominator. 

Having  first  obtained  the  least  common  multiple  of  all  the  denomi- 
nators of  the  given  fractions,  we  assume  this  to  be  their  least  common 
denominator.  We  then  take  such  a  part  of  it  as  is  expressed  by  each 
of  the  fractions  separately  for  their  respective  new  numerators.  Thus, 
to  get  a  new  numerator  for  |,  we  take  f  of  12,  the  least  common 
denominator,  by  dividing  it  by  3,  and  multiplying  the  quotient  4  by  2. 
We  proceed  in  like  manner  with  each  of  the  fractions,  and  write  the 
numerators  thus  obtained  over  the  least  common  denominator. 

In  this  process  the  value  of  each  fraction  remains  unchanged,  as 
both  terms  are  multiplied  by  the  same  number.     (Ai-t.  140.) 

Rule.  —  1.  Find  the  least  common  multiple  of  the  denominators  for  the 
least  common  denominator. 

2.  Divide  the  least  common  denominator  hy  each  given  denominator, 
and  multiidy  the  quotient  hy  the  corresponding  numerator,  for  the  new 
numerators. 

Note.  —  Compound  fractions  must  be  reduced  to  simple  ones,  whole  and 

141.  How  do  you  find  the  least  common  denominator  of  two  or  more  frac- 
tions ?  Upon  what  principle  does  this  process  depend  ?  What  is  the  rule 
for  reducing  fractions  to  their  least  common  denominator  1  What  must  bo 
done  with  compeund  fractions,  whole  numbers,  uud  mixed  uumbers  \ 


148  COMMON   FRACTIONS. 

mixed  numbers  to  improper  fractions,  and  all  to  their  lowjest  terms  before 
finding  tlie  least  common  denominator. 

Examples  for  Practice. 

2.  Reduce  f ,  f ,  f ,  and  ^  to  the  least  common  denominator. 

Ans   —9-0^   -AS^  J.GO    J.  0  5 

-tills.    i2(T'    12X7'    12ff)    iLjO"' 

3.  Reduce  f ,  f ,  f,  and  -^^  to  the  least  common  denominator. 

All-5     J485        792         880         360 
-'*^°"   l^BU'  TUS(T5  T?8(J5  T9F(T' 

4.  Reduce  |-,  -j^^j,  and  7f  to  the  least  common  denominator. 

An>J      35      36      310 

5.  Reduce  f,  y\,  ^i,  and  5f -to  the  least  common  denominator. 

Ans.  J-4  J-l  J-A  J„5  2. 

6.  Reduce  ^,  f ,  f,  f ,  f ,  and  -^^  to  the  least  common  denomi- 

•nnfnr  "     An<!     12     18     20      15     2  1     j.0 

7.  Reduce  ^,  §,  J,  :^,  ^,  and  -^^j  to  the  least  common  denomi- 
nator. Ans.  ^|,  f I,  if,  ^9^,  ^6^,  ^3^. 

8.  Reduce  ^,  f ,  and  -^^  to  the  least  common  denominator. 

Ans     so      16     2  1 

9.  Reduce  7-|,  5^^?  ''^j  ^"^^  ^  ^o  the  least  common  denominator. 

Ana     341    544      308      3F-2 

10.  Reduce  f,  4,  5,  7,  and  9  to  the  least  common  denomi» 
nator.  Ans.  f ,  JjS  -\^-,  -V-,  -=V^-. 

ADDITION. 

142.  Addition  of  Fractions  is  the  process  of  finding  the  sum  of 
two  or  more  fractions. 

Fractions  can  only  be  added  when  expressing  fractional  units 
of  the  same  kind. 

143,  To  add  fractions  having  a  common  denominator. 
Ex.  1.  Add  I,  f,  f,  I,  and  f  Ans.  2f. 

OPERATION.  '     The     fractions     all     being 

12.4,5.6  ^.        sevenths,  we  add  their  numer- 

^"T'_^"T";^~ry~ry==  \'  ^^  "T      ators,  and  write  their  sum,  18, 

over  the  common  denominator, 
7  ;  and  thus  obtain  ^  =  2^,  the  required  sum.     That  is,  we 

Write  the  sum  of  the  numerators  over  the  common  denominator. 


142.  What  is  addition  of  fractions  ?  —  14.3.  How  arc  fractions  having  a 
common  denominator  added  f     Give  the  reason. 


ADDITION.  149 
Examples  for  Practice. 

.  ii-uu  YT^  rr>  TT>  Tr»  tt'  ""^  tt*  j\nb.  oyy. 

3.  Add  -^f,  -fV,  j\,  -^j,  and  -fl.  Ans.  2^^. 

4.  Add  j=V,  2\,  if,  and  1^.  Ans.  2 ^L. 

5.  Add  If,  -If,  If,  and  f|.  Ans.  2if . 

6.  Add  -iif,  iH,  and  -rVr-  Ans.  l^-^f. 

7.  Add  if-ff,  iff f,  and  -rf fx-  Ans.  Ifff f. 

144,    To  add  fractions  not  having  a  comm.on  denomi- 
nator. 

Ex.  1.  What  is  the  sum  of  |,  f ,  and  -/g-  ?  Ans.  1^|. 


0   8    12 


OPERATION. 

2  4,  common  denominator. 


6 

8 

12 


4X5.: 
3X3  = 
2X7  = 

=  20 
=     9 
=  14 

new  numerators. 


Sum  of  numerators,  4  3 ,  ,  9     . 


2X2X2X3  =  24.    Cora,  denominator,    24         ^^' 

We  reduce  the  given  fractions  to  equivalent  ones  having  a  common 
denominator,  that  they  may  express  fractional  units  of  the  same  kind ; 
and  then  we  add  tlie  numerators,  and  write  their  sum  over  the  common 
denominator,  and  reduce  the  fraction. 

Rule.  —  Reduce  the  given  fractions  to  a  common  denominator.  Add 
the  numerators,  and  write  their  sum  over  the  common  denominator. 

Note  I.  —  First  reduce  mixed  numbers  to  improper  fractions,  and  com- 
pound fractions  to  simple  fractions,  and  each  fraction  to  its  lowest  terms. 

Note  2.  —  In  adding  mixed  numbers,  the  fractional  parts  may  be  added 
separately,  and  their  sum  added  to  the  amount  of  the  whole  numbers. 

Examples  for  Practice. 

2.  What  is  the  sum  of  f ,  f  i,  and  f  f  ? 

3.  What  is  the  sum  of  j^^,  \^,  and  -^^  ? 

4.  What  is  the  sum  of  ^f  and  f  f  ? 

5.  What  is  the  sum  o'f  f ,  |,  |,  and  -^  ? 

6.  Add  I,  /-J-,  a,  and  J-. 

7.  Add  fa,  5  J,  and  ^^. 

8.  Add  ^,,  l-g,  f  I,  and  ^%\y. 

144.  The  rule  for  adding  fractions  not  having  a  common  denominator  1 
How  may  mixed  numbers  he  added  ^ 
13* 


Ans 

.2i|. 

Ans.  1 

-t'ieV 

Ans. 

IHf 

Ans 

9  1 

Ans. 

nu- 

Ans,  1 

20  If 

Ans. 

^m- 

150  COMMON  FRACTIONS. 

9.  Add  i,  -2,  I,  I,  I,  f ,  and  I,  together.  Ans.  5^^^. 

in       AAA    8       9        10      li      12      13     orifl   J-4.  Ati<5     fil-4401 

11.  Add  I  of  f  to  f  of  §.                     •  Ans.  IJ-^. 


1 1 


12.  Add  f  of  ^  to  }i-  of  A.  Ans.  1 

13.  Add  ^  of  f  to  A  of  yV  -Ajis.  -fi^^. 

14.  Add  f  of  f  of  I  to  f  of  f  of  -/(J.  Ans.  j%. 

15.  Add  ^  of  y3y  of  II  to  }  of  f .  Ans.  ^^. 

16.  Add  3f  to  41^.  Ans.  8yV 

17.  Add  4f  to  5f.  Ans.  lOJ-J. 

18.  Add  17|  to  IByV  Ans.  36^. 

145i  To  add  two  fractions  having  1  for  their  numerator. 
Ex.  1.   Add  I  to  -i.  Ans.  ^^. 

OPEKATION.  We   first    find    the 

Sum  of  the  denominators,         4  -|-  5  =  9      product  of  the  denom- 

~A c        oTv      inators,   which   is    20, 

Product  of  the  denommators,  4  X  5  =  20     anxi  then    their   sum, 

■which  is  9,  and  write  the  former  for  the  denominator  of  the  required 
fraction,  and  the  latter  for  the  numerator. 

By  this  process  we  reduce  the  fractions  to  a  common  denominator, 
and  then  add  their  numerators.  Hence,  to  add  two  fractions  of  this 
kind, 

Write  the  sum  of  the  given  denominators  over  their  j)roduct. 

Examples  for  Practice. 
2.  Add  i  to  I,  I  to  i,  1  to  ^,  i  to  ^,  ^  to  ^. 

3.  Add  I  to   yV,  i  to   ^,  I  to   I,   i  to  yV,   i   to  yV  l  tO  i- 

4.  Add   i    to   I,    I    to   yV,    i   to   ^,  ^    to   yV,    -J    to   i     yV   tO  y^- 

5.  Add  \  to  -Jy,  ^  to  yV,  i  to  -^^j,  I  to  ^,  .1  to  |,  ^  to  f 

6.  Add  I  to  ^,  I  to  i,  I  to  i,  I  to  ^,  I  to  I,  I  to  yiy. 

7.    Add  i  to  I,   -I  to  i,   ^    to   ^,^    to    yV,   i   to   yV,   i   to  y^. 

SUBTRACTION. 

U6.    Suhtraclion  of  Fradions  is  the  process  of  finding  the  dif- 
ference between  two  fractions. 

Note.  —  One  fraction  can  he  subtracted  from  another  only  when  both 
express  fractional  units  of  the  same  kind. 

145.  How  can  vou  add  two  fractions  when  the  numerators  are  a  unitl 
The  reason  for  tliis  7  —  146.  What  is  subtraction  of  fractious  7 


SUBTRACTION.  151 

147.  To  subtract  fractions  having  a  common  denomi- 
nator. 

Ex.  1.  From  ^  take  f.  Ans.  |. 

OPERATION.         The  fractions  both  being  ninths,  we  subtract  the  less 
|.  —  f  =  f     numerator  from  the  greater,  and  write  the  difference,  5, 
over  the  common  denominator,  9  ;  and  thus  obtain  |^  as 
the  required  difierence.     That  is,  we 

W7'ite  the  difference  of  their  numerators  over  the  common  denominator. 
EXAJVIPLES    FOR    PRACTICE. 

2.  From  y^y  take  ^y. 

3.  From  \\  take  y%-. 

4.  From  f  f  take  4j. 

5.  From  ^\  take  Jyx- 

6.  From  y%^g-  take  J/gV 

7.  From  /jj  take  -^jj. 

8.  From  y^g-  take  -^-^^. 

148.  To  subtract  fractions  not  having  a  common  de- 
nominator. 

Ex.  1.  From  ||  take  y^.  Ans.  ^\. 


Ans. 

tV 

Ans. 

iV 

Ans. 

23 

Try 

Ans.  ■ 

148 
7  11* 

Ans. 

239 
tt64' 

Ans. 

tV 

Ans.  f . 

16     12 


1  6 
12 


OPERATION. 

4  8,  common  denominator. 


3  X  13  =  39  r 

^  new  numerators. 


=  39) 

4  X     7  =  28j 

4  X  4  X  3  =  48.  TT   ,.«.  n 

^      ^  11,  difference  of  numerators. 

4  8,  common  denominator. 

We  reduce  the  givei\  fractions  to  equivalent  ones  having  a  common 
denominator,  that  they  may  express  fractional  units  of  the  same  kind, 
and  then  we  subtract  the  less  numerator  from  the  greater,  and  place 
the  difference  over  the  common  denominator. 

Rule.  —  Reduce  the  fractions  to  a  common  denominator,  then  write 
the  difference  of  the  numerators  over  the  common  denominator. 

Note  —  If  the  minuend  or  subtrahend,  or  both,  are  compound  fractions, 
they  must  be  reduced  to  simple  ones. 


147.  How  do  you  subtract  fractions  havinp;  a  comi{Jon  denominator  ? - 
148.  The  rule  for  subtracting;  fractions  not  bavino;  a  common  denominator 
If  the  minuend  or  subtrahend  is  a  compound  fraction,  what  must  be  done  1 


i 


Ans. 

m 

Ans.  1^ 

Ans. 

t¥(7 

Ans.  II 

Ans. 

rVr 

Ans. 

4Vt 

Ans.  ^tU 

-A-ns.  -3-§§^ 

Ans. 

-lh\ 

Ans. 

-r^TT 

152  COMMON  FRA0TIONS. 

Examples  for  Practice. 

2.  From  y^g-  take  ^t-. 

3.  From  i9-  take  -f^-. 

4.  From  ^1^  take  /^. 

5.  From  -J^  take  y^- 

6.  From  f  ^  take  y^. 

7.  From  ^f  take  ^y. 

8.  From  ^^^  take  y^- 

9.  From  J^-  take  xjyVTy 

10.  From  f  of  y^y  take  A  of  f. 

11.  From  i  of  y^^y  take  y^  of  -ff. 

12.  From  |  of  12f  take  f  of  e^^^.  Ans.  |J. 

149.  To  subtract  a  proper  fraction  or  a  mixed  number 
from  a  whole  number. 

Ex.  1.  From  16  take  2f  Ans.  13f. 

OPERATION.  Since  we  have  no  fraction  from  which  to  sub- 

From  1  6  tract  the  ^,  we  add  1,  equal  to  f,  to  tlie  minu- 

Take       2^         end,  and  say  ^  from  |  leaves  f .     We  write  the 
■n         ,03         f  below  the  fine,  and  cany  1  to  the  2  in  the 
^         subtrahend,   and   subtract   as   in   subtraction  of 
simple  numbers. 

The  same  result  will  be  obtained,  if  we 

Subtract  the  number  denoting  the  numerator  from  that  denoting  the  de- 
nominator, and  under  the  remainder  write  the  denominator,  and  carry  1  to 
the  integral  part  of  the  subtrahend  before  subtracting  it  from  the  minuend. 

Note.  —  When  tho  subtrahend  is  a  mixed  nnmber,  we  may  reduce  it  to 
an  improper  fraction,  and  clianj^e  the  whole  number  in  the  minuend  to  a 
fraction  having  the  same  denominator,  and  then  proceed  as  in  Art.  148. 

Examples  for  Practice. 


2. 

3. 

4. 

5. 

6. 

From  1  2 

19 

13 

14 

17 

Take       4J 

H 

9iV 

8f 

6H 

Ans.       7^ 

15f 

m 

^ 

lOyV 

7.  From  23  take 

13^. 

Ans.  9§. 

8.  From  47  take 

hi 

Ans.  4G^jy. 

9.  From  139  tak 

%. 

c  7r4^. 

Ans.  C3^|. 

149.  How  do  you  suhtract  a  pioper  fraction  or  mixed  number  from  a 
whole  number  1     The  reason  for  this  rule  1 


SUBTRACTION.    '  153 

.     150.   To  subtract  one  mixed  number  from  another. 
Ex.  1.  From  9f  take  3f.  Ans.  5f |. 

Z^^^^  0  2  J-  Qio  ^6  fi^'^*-  reduce  the  fractional  parts  to  a 

T  b"^  ol  H  off  common  denominator  by  multiplying  the 
lake  o^       oga        terms  of  the  fraction  f  by  5,  the  denominator 

Rem.  5§|       of  the  other,  thus:  ^^g~ 3^  ;    and   then    the 

terms  of  the  fraction  f  by  7,  the  denominator  of  the  first,  thus : 
F^-Zl^-  Now,  since  we  cannot  take  f|  from  ^a,  we  add  1, 
equal  to  |f,  to  the  ^4  in  the  minuend,  and  obtain  |4,  We  next 
subtract  f  ^  from  f  |,  and  write  the  remainder,  f  *,  below,  and 
carry  1  to  the  3  in  the  sul)trahend,  and  subtract  from  the  9 
above,  as  in  simple  whole  numbers. 

SECOND    OPERATION. 

From  9f  =  ^J"  ==  -%^^^-  In  this  operation,  we  reduce 

Take   3f  =  ^-^  =  ^^^  the     mixed     numbers     to    im- 

x>  J^9  r 24       proper  fractions,  and  these  frac- 

^^  ^^      tions  to  a  common  denominator. 

"We  then  subtract  the  less  fraction  from  the  T^reater,  and,  reducing 
the  remainder  to  a  mixed  number  obtain  5f  |,  as  before.  Hence, 
in  pei-forming  like  examples, 

Reduce  the  frncdonal  parts,  if  necessary,  to  a  common  denominator, 
and  subtract  iJie  fractional  part  of  the  subtrahend  from  that  of  the  minu- 
end, as  in  Ai't.  147  ;  remembering  to  increase  the  fractional  part  of  the 
minuend,  when  otherwise  it  would  he  less  than  that  of  the  subtrahend,  be- 
fore subtracting,  by  as  many  fractional  units  as  it  takes  to  make  a  uriit  of 
the  fraction  (Art.  131),  and  carry  1  to  the  whole  number  of  the  subtrahend 
before  subtracting  it.     Or, 

Reduce  the  mixed  numbers  to  improper  fractions,  then  to  a  common  de- 
nominator, and  subtract  the.less  fraction  from  the  greater. 

Examples  for  Practice. 

2.  3.  4.  5.  6. 

From  9^  7i  8f  9^  lOJ 

TakeSi^  3^  4^  3|  10^^ 


Ans      32.3  9  17  Q22  KS  &a 

.tills.      O^y  Og-g-  Oj^  Jg-  f-Q 

150.  How  do  yoa  reduce  the  fractions  of  the  mixed  numbers  to  a  common 
denominator  ?  How  does  it  appear  that  this  process  reduces  them  to  a  com- 
mon denominator  ?  How  do  you  then  proceed  ?  What  other  metliod  of 
subtracting  mixed  numbci-s  1    How  may  all  like  examples  be  performed  ? 


154  COMMON   FRACTIONS.  ' 


7. 

8. 

From  1  2f 

16A 

Take      9^ 

5f 

Ans.       2|f 

lO^f 

9. 

10.                 11. 

19f 

9  7i               8  7^^ 

15f 

1  8/x             1  9t 

m 

7  8|f             6  7f  i 

Ans.  11||. 

Ans.  6||. 

Ans.  bJfj. 

Ans.  57ft. 

Ans.  27^^i. 

12.  From  19^  take  1-^^. 

13.  From  lb\  take  8^. 

14.  From  9yVy  take  3||. 

15.  From  71yL  take  IS/j. 

16.  Frem  61||  take  33|f. 

17.  From  a  hogshead  of  wine  there  leaked  out  12f  gallons; 
how  much  remained?  Ans.  50|  gallons. 

18.  From  $  10,  $  2|-  were  given  to  Benjamin,  $3^  to  Lydia, 
$  1 J^  to  Emily,  and  the  remainder  to  Betsey ;  what  did  she  re- 
ceive ?  Ans.  $  3^. 

151.  To  subtract  one  fraction  from  another,  when  both 
have  1  for  their  numerator. 

Ex.  1.  Wliat  is  the  difference  between  ^  and  \  ?      Ans.  ^j. 

OPERATION. 

Difference  of  the  denominators,  7  —  3  =  4 
Product  of  the  denominators;,      7  X  3  =  21 

We  first  find  the  product  of  the  denominators,  wliich  is  21,  and  then 
their  difference,  which  is  4,  and  write  tlie  former  for  the  denominator 
of  the  required  fraction,  and  the  latter  for  the  numerator.  By  this 
process  the  fractions  are  reduced  to  a  common  denominator,  and  their 
difference  found.  Hence,  to  find  the  difference  of  two  fractions  of 
this  kind. 

Write  the  difference  of  the  denominators  over  their  product. 

Examples  for  Practice. 

2.  Take  \  from  -J,  \  from  ^,  ^  from  ^,  f  from  ^. 

3.  Take  ^  from  ^,  ^  from  |,  \  from  ^,  \  from  ^. 

4.  Take  ^  from  ^,  f  from  ^,  -^  from  ^,  -^  from  ^. 

5.  Take  ^  from  |,  ^  from  |,  -^  frem  ^,  \  from  \. 

6.  Take  |  from  ^,  ^  from  \,  -^  from  ^,  ^  from  ^. 

7.  Take  \  from  ^,  -J-  from  ^,  ^  from  ^,  yV  ^''o™  i- 

151.  How  do  you  subtract  one  fraction  from  another  when  both  fractiona 
have  a  unit  for  a  numerator  ?     What  is  the  reason  for  this  urocess  1 


MULTIPLICATION.  155 

MULTIPLICATION. 

152i  Multiplication  of  Fractions  is  the  process  of  taking  one 
number  as  many  times  as  there  are  units  in  another,  when  one 
or  both  of  the  numbers  are  fractions. 

153,  To  multiply  a  fraction  by  a  whole  number. 

Ex.  1,   MuUiply  ^  bj  4.  Ans.  3^. 

FIRST  OPERATION.  Jn  thc  first  operation  we  multiply  the 

§•  X  4  =  -g^-  =  3J-  numerator  of  the  fraction  by  the  whole 
number,  and  obtain  3^  for  the  answer. 
It  is  evident  that  the  fraction  ^  is  multiplied  by  multiplying  its 
numerator  by  4,  since  the  parts  taken  are  4  times  as  many  as 
before,  while  the  parts  into  which  the  number  or  thing  is  divided 
remain  the  same.     Therefore, 

Multiplying  the  numerator  of  a  fraction  hy  any  number  multi- 
plies the  fraction  by  that  number. 

SECOND  OPERATION.  In  thc  second  operation  we  divide  the 

1^  X  4  =  J  ==  3^  denominator  of  the  fraction  by  the  whole 

number,  and  obtain  3 J-  for  the  answer, 
as  before.  It  is  evident,  also,  that  the  fraction  |-  is  multiplied  by 
dividing  its  denominator  by  4,  since  the  parts  into  which  the 
number  or  thing  is  divided  are  only  \  as  many,  and  conse- 
quently 4  times  as  large,  as  before,  while  the  parts  taken  remain 
the  same.     Therefore, 

Dividing  the  denominator  of  a  fraction  by  any  number  multi- 
plies the  fraction  by  that  number. 

Rule.  —  Multiply  the  numerator  of  the  fraction  hy  the  whole  number. 
Or, 

Divide' the  denominator  of  the  fraction  hy  the  whole  nuinber,  when  it  can 
he  done  without  a  remainder. 

ExAjiPLES  FOR  Practice. 

2.  Multiply  f  by  9.  Ans.  6f . 

3.  Multiply  T-\  by  5.  Ans.  2|. 

4.  Multiply  ^6  by  3.  Ans.  If 

5.  Multiply  If  by  85.  Ans.  49. 


152.  What  is  multiplication  of  fractions  ?  —  153.  How  is  a  fraction  multi- 
plied, by  the  first  operation  ■?  The  reason  of  the  operation  1  What  inference 
is  drawn  from  itl  How  is  a  fraction  multiplied,  by  the  second  operation? 
The  reason  of  the  operation  1  What  inference  is  drawn  from  it  1  The  rule 
for  multiplying  a  fraction  by  a  whole  number  1 


156  COXLMON   FRACTIONS. 

6.  Multiply  J-i  by  83.  Ans.  76yV- 

7.  Multiply  If  by  189.  Ans.  lG6|f 

8.  Multiply  iif  by  365.  Ans.  352/tV 

9.  Multiply  f  ^  by  48.  Ans.  43^. 

10.  If  a  man  receive  f  of  a  dollar  for  one  day's  labor,  what 
will  he  receive  for  21  days'  labor  ?  Ans.  $  7^. 

11.  "What  cost  561b.  of  chalk  at  f  of  a  cent  per  lb.? 

Ans.  $  0.42. 

12.  "What  cost  3961b.  of  copperas  at  -^y  o^  ^  ^^^^  P^^  ^^-  ^ 

Ans.  $  3.24. 

13.  "What  cost  79  bushels  of  salt  at  |  of  a  dollar  per  bushel? 

Ans.  $  69f 

154.   To  multiply  a  whole  number  by  a  fraction. 
Ex.  1.   Multiply  15  by  f.  Ans.  9. 

FIRST  OPERATION.         In  the  first  operation  we  divide  the  ichole 
5)15  member  by  the  denominator  of  the   fraction, 

~      „ „      and  obtain  ^  of  it.     "We  then  multiply  this 

^  quotient  by  3,  the  numei-ator  of  the  fraction, 

and  thus  obtain  %  of  it,  which  is  9. 

SECOND  OPERATION.       In  thc  sccoud  operation  we  multiply  the 

1  5  wJiole  number  by  the  numerator  of  the  frac- 

3  tion,  and  divide  the  pi'oduct  by  the  denomi- 

4^ -i-  5  =  9       nator,  and  obtain  9  for  the  answer,  as  before. 

Therefore, 

* 
Multiplying  by  a  fraction  is  talcing  the  part  of  the  multiplicand 

denoted  by  the  multiplier. 

Rule.  —  Divide  the  tvJiole  number  by  the  denominator  of  the  fraction, 
when  it  can  be  done  tmthout  a  remainder,  and  multiply  the  quotient  by  the 
numerator.     Or, 

Multiply  the  whole  number  by  the  numerator  of  the  fraction,  and  divide 
the  product  by  the  denominator. 

Examples  for  Practice. 

2.  Multiply  36  by  |.  Ans.  28. 

3.  Multiply  144  by  \\.  A.ns.  88. 

4.  Multiply  375  by -|-|.  Ans.  325. 

5.  Multiply  2-277  by  \%.  Ans.  1610. 

6.  Multiply  376  by  ||.  Ans.  243tV- 

154.  How  do  you  tnulli]ily  a  wliolo  iniiiibcr  by  a  fnu-tion,  accordinp:  to  tho 
first  opiTution  ?  Hdw  by  tlic  sccdiid?  Wliat  iiifiTcnce  is  drawn  from  tlie 
operation  1     The  rule  for  multiplying  a  wliole  number  by  a  fraction  J 


MULTIPLICATION.  157 

7.  Multiply  471  by  ^-f^.  Ans.  8/g-. 

8.  Multiply  871  by  ^V-  A»s.  23f^. 

9.  Multiply  867  by  -^^^.  Ans.  6-rV^. 

155,    To  multiply  a  whole  and  mixed  number  together. 
Ex.  1.  Multiply  17  by  6f.  Ans.  114f. 

OPERATION. 

17 

g  i  We  first  multiply  17   by  6,  the  whole 

—  number  of  the  multiplier,  and  then  by  the 

^  ^  ''^  fractional  part,  f ,  which   is   simply  taking 

J  or  1  7  =     12-^  1^  of  it ;  and  add  the  two  products. 
114| 

Ex.  2.   Multiply  7f  by  4.  Ans.  30|. 

OPERATION.  "We   first    multiply  f  in  the  multiplicand 

7  f         by  4,  the  multiplier  ;  thus,  4  times  f  are  -^■^-, 

4  equal  to  2f,  which  is  in  effect  taking  f  of  the 

3  of  4  =      2^         multiplier,  4.     We  then  multiply  the  whole 

''  2  8^  number,  7,  by  4 ;  and  add  the  two  products. 

Hence,  in  performing  like  examples, 


30| 

Multiply  the  fractional  part  and  the  whole  number  separately, 
and  add  the  products. 

Examples  for  Practice. 

3.  Multiply  91  by  5.  Ans.  46|. 

4.  Multiply  12f  by  7.  ■  Ans.  88^. 

5.  Multiply  9  by  S}^.  Ans.  80f 

6.  Multiply  10  by  7f  Ans.  71^. 

7.  Multiply  11 6  by  8.  Ans.  94f 

8.  What  cost  7y\lb.  of  beef  at  5  cents  per  pound  ? 

Ans.  $  O.37-J83-. 

9.  What  cost  2337jbbl.  of  flour  at  $  6  per  barrel  ? 

Ans.  $  141 4-. 

10.  What  cost  8fyd.'  of  cloth  at  $  5  per  yard  ? 

Ans.  $41  J. 

11.  What  cost  9  barrels  of  vinegar  at  $  6f  per  barrel.'' 

Ans.  $  57f . 


155.  The  nile  for  multiplyiijg  a  whole  and  mixed  number  together?   Does 
it  make  any  difference  which  is  taken  for  the  multiplier  ? 
14 


158 


COMMON  FRACTIONS. 


12.  What  cost  12  cords  of  wood  at  §  6.37^  per  cord? 

Alls.  $  76.50. 

13.  What  cost  llcwt.  of  sugar  at  %  9f  per  cwt.  ? 

Ans.  $  103^ 

14.  What  cost  4f  bushels  of  rye  at  $  1.75  per  bushel  ? 

Ans.  $  7.6o|. 

15.  What  cost  7  tons  of  hay  at  $  llf  per  ton  ? 

Ans.  $  83f 

16.  What  cost  9  doz.  of  adzes  at  $  10|  per  doz.  ? 

Ans.  $  95|. 

17.  What  cost  5  tons  of  timber  at  $  3^  per  ton  ? 

Ans.  $  15|. 

18.  What  cost  I5cwt.  of  rice  at  $  7.624  per  cwt.  ? 

Ans.  $  114.37|. 

19.  What  cost  40  tons  of  coal  at  $  8.37^  per  ton  ? 

Ans.  $  335. 

156.   To  multiply  a  fraction  by  a  fraction. 

Ex.  1.   Multiply  f  by  f.  Ans.  -/j. 


OPERATION   BY   CANCELLATION. 


OPERATION. 


^x 


f^ 


tV 


7       0 

0X1 

3 


7 


To  multiply  |^  by  f  is  to  take  |  of  the  multiplicand,  |-  (Art 
154).  Now,  to  obtain  f  of  ^,  we  simply  multiply  the  numerators 
together  for  a  new  numerator,  and  the  denominators  together 
for  a  new  denominator  (Art.  138).     Therefore, 

Multiplying  one  fraction  by  another  is  the  same  as  reducing 
compou7id  fractions  to  simple  ones. 

Rule.  —  ISIuliiphj  the  numerators  together  for  a  new  numerator,  and 
the  denominators  together  for  a  new  denominator. 

Note.  —  When  there  arc  factors  common  to  tlic  numerators  and  denomi- 
nators, the  work  may  be  shortened  by  canceling  those  factors. 


Examples  for  Practice. 


2.  Multiply  I  by  ^V 

3.  Multiply  ^  by  ^^. 


Ans.  -^Y* 
Ans.  ^. 


156.  What  is  llic  first  rule  for  ninlti])lyinp;  one  fraction  by  another?  How 
docs  it  a])pcar  tluit  this  operation  multiplies  tiie  fraction  of  the  multiplicand? 
What  is  tiie  inference  driiwn  from  it  ?     What  Is  the  note  ? 


MULTIPLICATION.  159 

4.  Multiply  f^  by  ^f .  Ans.  4. 

5.  Multiply  if  by  i§.  Ans.  ■^. 

6.  Multiply  if  by  ^J.  Ans.  ^. 

7.  Multiply  I  by  ^V  Ans.  ^f  j. 

8.  Multiply  ^%  by  §^.  Ans.'^. 

9.  What  cost  ^  of  a  bushel  of  com  at  f  of  a  dollar  per  bushel  ? 

Ans.  ^  of  a  dollar. 

10.  If  a  man  travels  -j\  of  a  mile  in  an  hour,  how  far  would 
he  travel  in  ^  J  of  an  hour  ?  Ans.  :|-  of  a  mile. 

11.  If  a  bushel  of  corn  will  buy  y^j  of  a  bushel  of  salt,  how 
much  salt  might  be  bought  for  ^  of  a  bushel  of  corn  ? 

Ans.  f-^  of  a  bushel. 

12.  If  §  of  f  of  a  dollar  buy  one  bushel  of  corn,  what  will  ^ 
of  y\  of  a  bushel  cost  ?  Ans.  -^-^  of  a  dollar. 

13.  If  f  of  f  of  -^Y  of  an  acre  of  land  cost  one  dollar,  how 
much  may  be  bought  with  f  of  $  18  ?  Ans.  Iff  acres. 

157i ,  To  multiply  a  mixed  number  by  a  mixed  number. 
Reduce    them    to    improper   fractions,    and    then   proceed    as    in 


Art.  156. 

Ex.  1.   Multiply  4f  by  6§. 

Ans.  30|. 

OPERATION. 

4f  =  ^4;6f  =  ^. 
4 

23^^0        92       _ 
$""     3-3-       ''i 

Examples  for  Practice. 

2.  Multiply  7i  by  8f                                   "           Ans.  6O3V 

3.  Multiply  Al  by  9^.                                              Ans.  45/^. 

4.  Multiply  llf  by  8f                                            Ans.  99^f 

5.  Multiply  12f  by  llf.                                            Ans.  147^. 

6.  What  cost  11  cords  of  wood  at  $  5f  per  cord  ? 

Ans.  $4124-. 

7.  What  cost  7|yd.  of  cloth  at  $  3^  per  yard  ?   Ans.  $  25|f . 

8.  What  cost  Gf  gallons  of  molasses  at  23|^  cents  per  gallon  ? 

Ans.  $  1.52^|. 

9.  If  a  man  travel  3|  miles  in  one  hour,  how  far  will  he  travel 
in  9^  hours  ?  Ans.  34=ig-. 

157.  How  do  you  multiply  a  mixed  number  by  a  mixed  number  ? 


160  COMMON  FRACTIONS. 

iO.  What  cost  SGl^i  acres  of  land  at  $  25 1  per  acre? 

Aus.  $9167^^§. 

11.  How  many  square  rods  of  land  in  a  garden,  which  is  97^% 
rods  long,  and  49f  rods  wide  ?  Aus.  4810^^  rods. 

DIVISION. 

158.  Division  of  Fractions  is  the  process  of  dividing  when  the 
divisor  or  dividend,  or  both,  are  fractions. 

159.  To  divide  a  fraction  by  a  whole  number. 

Ex.  1.    Divide  |  by  4.  Ans.  f. 

FIRST  OPERATION.  Wc  divide  the  numerator  of  the  fi-action  by  4, 

g  2         ^^^  write  the  quotient,  2,  over  the  denominator. 

— 1_  4  __  _  It   is   evident  this  process   divides  the   fraction 

9  9         by  4,  since   the   size  of  the  parts  into  which  the 

whole  number  is  divided,  as  denoted  by  the  denom- 
inator, remains  the  same,  while  the  number  of  parts  taken  is  only  ^  as 
many  as  before.     Therefore, 

Dividing  the  numerator  of  a  fraction  hy  any  number  divides  the  fraction 
by  that  number. 

Ex.  2.    Divide  f  by  9.  Ans.  ■^^. 

SECOND  OPERATION.        We  multiply  the  denominator  of  the  fraction  by 
K  K        the  divisor,  9,  and  write  the  product  under  the 

J-  9  =  —       numerator,  5. 

«■  63  It  is  evident  this  process  divides  the  fraction, 

since  multiplying  the  denominator  by  9  makes 
the  number  of  parts  into  which  the  whole  number  is  divided  9  times 
as  many  as  before,  and  consequently  each  part  can  but  have  \  of 
its  fjrmer  value.  Now,  if  each  part  has  but  \  of  its  former  value, 
while  only  the  same  number  of  parts  is  expressed  by  the  fraction, 
it  is  plain  the  fraction  has  been  divided  by  9.     Therefore, 

Multiplying  the  denominator  of  a  fraction  by  any  number  divides 
the  fract,ion  by  that  number. 

Rule.  —  Divide  the  numerator  of  the  fraction  by  the  whole  number, 
when  it  can  be  done  without  a  remainder,  and  ivrite  the  quotient  over  the 
denominator.     Or, 

Multiply  the  denominator  of  the  fraction  by  the  whole  number,  and  write 
tlie  product  under  the  numerator. 

158.  What  is  division  of  common  fractions?  — 159.  How  is  the,  fraciion 
divided  by  the  first  openition  ?  What  iiiforence  may  be  <lra\vn  from  this 
operation  7  How  is  a  fraction  divided  hy  tiic  second  operation'?  What 
inference  is  drawn  from  tliis  operjiiion  ?     The  rule  ? 


DIVISION.  161 

ExAjrpLEs  FOR  Practice. 


3.  Divide  -^^  by  3.  Ans.  ^tj. 

4.  Divide  |f  by  6,  Ans.  y-g-. 


7.  Divide  |J  by  9.  ■•  Ans.  jg-. 

8.  Divide  ^|^  by  15.  Ans.  ^-^. 


10.  Divide  §  by  12.  Ans. 


2 

3 

5.  Divide  y^  by  12.  Ans.  yj^. 

6.  Divide  -j-J-  by  8.  -Ajis.  ^^. 

3 

9.  Divide  ^f ^  by  75.  Ans.  -^f^. 

•  7 
TO"F' 


11.  John  Jones  owns  f  of  a  share  in  a  railroad,  valued  at 
$  117  ;  this  he  bequeaths  to  his  five  children.  What  part  of  a 
shai'e  wiU  each  receive  ?  Ans.  ^. 

•   12.  Divide  ^%  by  15.  Ans.  ^f^. 

13.  Divide  ■^%  by  28.  Ans.  5-f  j. 

14.  James  Page's  estate  is  valued  at  $  10,000,  and  he  has 
given  f  of  it  to  the  Seamen's  Society;  J  of  the  remainder  he 
gave  to  his  good  minister;  and  the  remainder  lie  divided  equally 
among  his  4  sons  and  3  daughters.  What  sum  will  each  of  his 
children  receive  ?  Ans.  $  680^^'^. 

160.   To  divide  a  whole  number  by  a  fraction. 

Ex.  1.    How  many  times  wiU  13  contain  ^?  Ans.  30^. 

OPERATION. 

7  3  3 

13  will  contain  ^  as  many  times  as  there  are  sevenths  in  13,  equal 
91  sevenths.  Now,  if  13  contain  1  seventh  91  times,  it  will  contain 
|-  as  many  times  as  91  will  contain  3,  or  301 

Rule.  —  Multiply  the  whole  number  by  the  denominator  of  the  fraction^ 
and  divide  the  product  by  the  numerator. 

Examples  eor  Practice. 

2.  Divide  18  by  |.  •  Ans.  20f. 

3.  Divide  27  by  J-i.  '  Ans.  29tV 

4.  Divide  23  by  |.  Ans.  92. 

160.  The  rale  for  dividing  a  whole  number  by  a  fraction  1  The  reason  for 
the  rule  1 

14* 


162  COMMON  FRACTIONS. 

5.  Divide  5  by  |.  Ans.  25. 

6.  Divide  12  by  £.  Ans.  16. 

7.  Divide  16  by  ^.  Ans.  32. 

8.  Divide  100  by  j^.  Ans.  lll|f. 

9.  I  have  50  square  yards  of  cloth ;  how  many  yards,  f  of  a 
yard  wide,  will  be  sufficient  to  line  it?  Ans.  83^  yards. 

10.  A.  Poor  can  walk  3^^^  miles  in  60  minutes  ;  Benjamin  can 
walk  y\  as  fast  as  Poor.  How  long  will  it  take  Benjamin  to 
walk  the  same  distance  ?  Ans.  73^*minutes. 

161.   To  divide  a  mixed  number  by  a  whole  number. 
Ex.  1.   Divide  17f  by  6.  Ans.  2f  f. 

OPEKATION.  Having    divided    the    whole 

6)  1  7y  number  as  in  simple  division,  we 

2        53__43..i3       __i§.        have  a  remainder  of  5|,  which 

)  F         s^  '    8  X  6       48  '        we  reduce  to  an  improper  frac- 

2  -|-  |-f  =  2|^§.  tion,  and  di^^de  it  by  the  divisor, 

as  in  Art.  159.     Annexing  this 
result  to  the  quotient  2,  we  obtain  2||  for  the  answer.     That  is,  we 

Divide  the  integral  part  of  the  mixed  number ;  and  the  remainder,  re- 
duced if  necessary  to  a  simple  fraction,  divide  as  is  in  Art.  159. 

Examples  for  Practice. 

2.  Divide  17|  by  7.  Ans.  2^." 

3.  Divide  18f  by  8.  Ans.  2^1. 

4.  Divide  27||  by  9.  Ans.  3-^^. 

5.  Divide  31Jiy  by  11.  Ans.  2-iVo- 

6.  Divide  78f  by  12.  Ans.  6fJ. 


7.  Divide  189||  by  4.  Ans.  47^ 

8.  Divide  107 ^V  ^7  ^-  Ans.  35§ 


3 
(J* 

ITS- 


9.  Divide  $  14f  among  7  men.  Ans.  $  2^^. 

10.  Divide  $  106J  among  8  boys.  Ans.  $  13f ^. 

11.  What  is  the  value  of  ^  of  a  dollar  ?        Ans.  $  0.34ii. 

12.  Divide  $  107j\  among  4  boys  and  3  girls,  and  give  each 
of  the  girls  twice  as  much  as  each  of  tlic  boys? 

Ans.  Boy's  share,  $  10^  ;  Girl's  share,  21||. 

13.  If  $  14  will  purchase  ^J  of  a  ton  of  copperas,  what  quan- 
tity will  $1  purchase?  Ans.  If'jCwt. 


161.  How  do  you  divide  a  mixed  number  by  a  whole  number  1 


OP 

H 

5 

ERATION. 
25 

5 

23) 

125  (51 J 
115 

DIVISION.  163 

162.    To  divide  a  whole  number  by  a  mixed  number. 
Ex.  1.  Divide  25  by  4f.  Ans.  b^%. 

AVe  first  reduce  the  divisor  and  dividend  to  fifths, 
and  then  divide  as  in  whole  numbers. 

The   divisor   and  dividend  v?ere  both  multiplied 
by  the  same  number,  5  ;  therefore  their  relation   to 

each  other  is  the  same  as  belbre,  and  the  quotient 

■j^Q  is  not  changed.     (Art.  115,  Note.)     Hence, 

Reduce  the  divisor  and  dividend  to  the  same  fractional  parts  as  are 
denoted  by  the  denominator  of  the  fraction  in  the  divisor,  and  then  divide 
as  in  lohole  numbers. 

Examples  for  Practice. 

2.  Divide  36  by  9  J.  Ans.  ^^. 

3.  Divide  97  by  13Ai-.  Ans.  6J-ff . 

4.  Divide  113  by  21f  .           Ans.  5^Vb- 

5.  Divide  342  by  14^-^1.  Ans.  23^^ 

6.  Tiiere  is  a  board  1 9  feet  in  length,  which  I  wish  to  saw  into 
pieces  2f  feet  long ;  what  will  be  the  number  of  pieces,  and  how 
many  feet  will  remain  ?  Ans.  7|f  pieces. 

.  163,    To  divide  a  fraction  by  a  fraction. 
Ex.  1.  Divide  |-  by  f.  Ans.  If^. 

FIRST  OPERATION.                                                      SECOND  OPERATION. 
7X9 63.    63         _ — 63 ,3,  X-i-4 XVS. 63  131 

Since  1  is  contained  in  -^ ,  i  times,  ^  is  contained  in  |-  9  times  |- 
times,  or  ^  times ;  and  A  is  contained  in  -^ ,  ^  of  ^  times,  which  is 
If,  or  1|4,  times. 

That  is,  we  have  multiplied  the  denominator  of  the  dividend  by  the 
number  denoting  the  numerator  of  the  divisor,  and  the  numerator  of  • 
the  dividend  by  the  number  denoting  the  denominator  of  the  divisor  \ 
hence,  for  convenience,  as  in  the  second  operation,  we  can  simply  in- 
vert the  terms  of  the  divisor  and  proceed  as  in  Art.  156. 

■162.  How  do  you  divide  a  wholo,  by  <i  mixed  number?  How  does  it 
appear  that  this  process  does  not  alter  the  quotient?  — 163.  How  do  you 
divide  a  fraction  by  a  fraction  ?  Give  the  reason  why  this  process  divides 
the  fraction  of  the  dividend. 


164  COMMON  FRACTIONS. 

Rule.  —  Invert  the  divisor,  and  then  proceed  as  in  multiplication  of 
fractions. 

Note  1.  —  Factors  common  to  numerator  and  denominator  should  be 
canceled. 

Note  2.  —  When  the  divisor  and  dividend  have  a  common  denominator 
their  denominations  cancel  each  other,  and  the  division  may  be  performed  by 
simply  dividing  the  numerator  of  the  dividend  by  that  of  the  divisor. 

Examples  for  Practice. 


2. 

Divide 

Jbyf 

Ans.  ' 

Ifl 

3. 

Divide 

^byi.    • 

Ans. 

3^ 

4. 

Divide 

if  by  U- 

Ans. 

H 

5. 

Divide 

1  by  j\. 

Ans. 

2| 

6. 

Divide 

T%  by  |. 

Ans.  6j% 

7. 

Divide 

1  by  ^\. 

Ans. 

4| 

8. 

Divide 

Tu  by  gV 

'■ 

t 

Ans.  6 

9. 

Divide 

U  by  ^V 

Ans. 

2f 

10. 

Divide 

f  of^by^of 

f. 

Ans.  18f. 

11. 

Divide 

4of-6_of_7^1 

by 

§ 

of 

1 

off 

Ans, 

tV 

12. 

Divide 

f  of  1  of  f  by 

of 

6 
T 

of 

T^F. 

Ans. 

3f 

164i   To  divide  a  mixed  number  by  a  mixed  number. 
Reduce  them  to  improper  fractions,  and  proceed  as  in  Art.  163. 
Ex.  1.     Divide  7f  by  3f .  Ans.  2||. 


OPERATION. 


7i  =  V;  H  =  -^~''- 


¥X2^i  =  W-=2s^ 


Examples 

FOR  Practice. 

2.  Divide  7f  by  U. 

Ans.  Iff. 

3.  Divide  3^-  by  7^-. 

Ans.  -J^. 

4.  Divide  ll^by  5f. 

Ans.  2yy2-. 

5.  Divide  4^  by  If 

Ans.  2j^fj. 

6.  Divide  llGf  by  14f. 

Ans.  8f f. 

7.  Divide  Blf  by  9f 

Ans.  8if  f 

8.  Divide  |  of  5i  of  7  by  | 

of3YV 

Ans.  llf 

16.3  The  rule  for  dividing  one  fraction  by  another?  How  may  fractions 
be  divided  wlicn  they  hiivc  a  common  dcnoiniimtor  ?  Docs  this  process  differ 
in  principle  from  the  other?  —  164.  How  do  you  divide  a  mixed  number  by 
a  mixed  number  ? 


COMPLEX   FRACTIONS.  165 

COMPLEX  FRACTIONS. 

165.    To  reduce  complex  to  simple  fractions. 

Ex.  1.  Reduce  3  to  a  simple  fraction.  Ans.  ^j. 

OPERATION.  Since  the  numerator  of  a  fraction  is  the 

ff  _^  1    y  8  __   8       dividend,   and  the   denominator  the  divisor 
I        ^        -J       2T      (^^Y^^  132),  we  simply  divide  the  numerator, 

^,  by  the  denominator,  f ,  as  in  division  of  » 
fractions.     (Art.  163.) 

Ex.  2.  Reduce  j-  to  a  simple  fraction.  Ans.  1^. 

OPERATION.  "Vi^e  reduce  the  numerator, 

__'  I  __  8.  \y  2  _^  J  6.  =^  1  7  8,  and  the  denominator,  4^,  to 

4^        f        ^        ^        ^  ^  improper  fractions,  and  then 

proceed  as  in  Ex.  1. 

Ex.  3.  Reduce  ■      •  ^  to  a  simple  fraction.  Ans.  If. 

OPERATION.  "VVe  reduce  the   denomi- 

3  3' 

T  =1  =  3  K/  6__ifl_.j2  nator,  ^  of  f,  to  a  simple 
^  of  f       f  i  7   fraction^  and  then  proceed  as 

before. 

Rule.  —  Reduce  tlie  terms  of  the  complex  fraction,  if  necessary,  to  the 
form  of  a  simple  faction.  Then  divide  the  numerator  of  the  complex 
fraction  by  its  denominator.  » 

Note.  —  Another  method  is  to  multiply  both  terms  of  the  complex  fraction  by 
a  common  multiple  of  their  denominators. 

Examples  for  Practice. 

12 
4.  Reduce  --  to  a  whole  number.  Ans.  28. 

3 
5    Reduce  —  to  a  simple  fraction.  Ans.  /j. 

14 

6.  Reduce  -^  to  a  simple  fraction.  Ans.  ^f . 


165.  The  rule  for  reducing  complex  to  simple  fractions  1    How  does  this 
process  differ  from  division  of  fractions  1 


166  COMMON  FRACTIONS. 

3 

7.  Reduce.  —  to  a  simple  fraction.  Ans.  y^t- 

u 

5 

8.  Chan"-e  -^  to  a  simple  fraction.  Ans.  ^§. 

7f 

33 

9.  Change  ■—-  to  a  mixed  number.  Ans.  21^. 

"a 
93 

10.  Reduce    —  to  a  simple  fraction.  Ans.  ■^^-^. 

122- 

9X 

11.  If  7  is  the  denominator  of  the  following  fraction,  — *  ,  what 

°  '  12^ 

is  its  value  when  reduced  to  a  simple  fraction  ?  Ans.  y^^^r* 

8 

12.  If  |-  is  the  numerator  of  the  following  fraction,  ~,  what  is 
its  value  when  reduced  to  a  simple  fraction  ?  Ans.  f  J. 

166.  Complex  fractions,  after  being  reduced  to  simple  ones, 
may  be  added,  subtracted,  multiplied,  and  divided,  like  them. 


Examples  for  Practice. 

1. 

Add  "I  and  r^.  together. 
T          ^^2- 

Ans.  It^VV 

2. 

72.           7 
Add  ~-  and  —  together. 
T            tV 

Ans.  25^85. 

3. 

3             1 
From  ^  take  |. 

Ans.  /^VV  ^ 

4. 

From  ?i  take  1 

Ans.  8^f. 

5. 

Multiply  f  of  P^by^of^^g. 

Ans.  T-iir- 

6. 

3A        6^ 
Multiply  ^1  by  2|- 

Ans.  ItWj- 

7. 

Divide  -f- of  12^  by  ;^  of8f. 

TT                            'i 

Ans.  103f 

166.  How  do  you  add,  subtract,  multiply,  and  divide  complex  fractions  ? 


LEAST   COMMON   MULTIPLE.  167 


GREATEST   COMMON  DIVISOR  OF  FRACTIONS. 

1G7.  To  find  the  greatest  common  divisor  of  two  or 
more  fractions. 

Ex.  1.   "WTiat  is  the  greatest  common  divisor  of  f ,  §,  and  ^|  ? 

OPERATION. 

4     2      12   20      3a    5S. 

F'  ■J'  T"&    3"5'  T5'  4  5* 

Greatest  common  divisor  of  the  numei-ators  =  ^    ")  greatest   com- 

.  n    y      c        •  -— ^  f  moil  divisor  re- 

l/east  common  denomuiator  ot  the  tractions  =  4  a  j  quired. 

Having  reduced  the  fractions  to  equivalent  fractions  with  the  least 
common  denominator  (Art.  141),  we  iind  the  greatest  common  divisor 
of  the  numerators  20,  30,  and  36,  to  be  2.  (Art.  124.)  Now,  since 
20,  30,  and  36  are  forty-Jifths,  their  greatest  common  divisor  is  not  2, 
a  whole  number,  but  so  many  for hj-Jiflhs.  Therefore  we  write  the  2 
over  the  common  denominator  45,  and  have  ^^  as  the  answer. 

Rule.  —  Reduce  the  fractions,  if  necessary,  to  the  least  common  de- 
nominator. Then  find  the  greatest  common  divisor  of  the  numerators, 
which,  written  over  the  least  common  denominator,  will  give  the  greatest 
common  divisor  required. 

ExAJiPLEs  FOR  Practice. 
2.  What  is  the  greatest  common  divisor  of  ^,  #,  and  14-  ? 


o 


Ans.  2^' 


3.  What  is  the  greatest  common  divisor  of  -i^f ,  ^,  ^^y,  and  ^f  ? 


Ans.  j-y-j. 


4.  What  is  the  greatest  common  ditisor  of  -i-f »  2^,  4,  and  5^  ? 

Ans.  :jlg., 

5.  There  is  a  three-sided  lot,  of  which  one  side  is  ]:66|ft., 
another  side  156|-ft.,  and  the  third  side  208^ft.  What  must  be 
the  length  of  the  longest  rails  that  can  be  used  in  fencing  it, 
allowing  the  end  of  each  rail  to  lap  by  the  other  ^ft.,  and  all  the 
panels  to  be  of  equal  length  ?  Ans.  lO^^ft. 

LEAST   COMMON  MULTIPLE   OF   FRACTIONS. 

168i    To  find  the  least  common  multiple  of  fractions. 

Ex.  1.   What  is  the  least  common  multiple  of -^t,  1^,  and  5^  ? 

Ans".  -V-  =  10^. 

167.  The  rule  for  finding  the  greatest  common  divisor  of  fr.actions  ?  Why, 
in  the  operation,  was  the  divisor  2  written  over  the  denominator  45  ? 


168  COMMON  FRACTIONS. 

OPERATION. 
T2"J  ■■■#'  ^¥  —  ^'  t)    I  • 

Least  common  multiple  of  the  numerators      =21)  least      corn- 
Greatest  common  divisor  of  the  denominator  =  2    )  required. 

Having  reduced  the  fractions  to  their  lowest  terms,  we  find  the  least 
common  multiple  of  the  numerators,  1,  3,  and  21,  to  be  21.  (Art. 
128.)  Now,  since  the  1,  3,  and  21  are,  from  the  nature  of  a  fraction, 
dividends  of  which  their  respective  denominators,  6,  2,  and  4,  are  the 
divisors  (Art.  132),  the  least  common  multiple  of  the  fractions  is  not 
21,  a  whole  number,  but  so  many  fractional  parts  of  the  greatest  com- 
mon divisor  of  the  denominators.  This  common  divisor  we  find  to  be 
2,  which,  written  as  the  denominator  of  the  21,  gives  ^l-  =  10^  as  the 
least  number  that  can  be  exactly  divided  by  the  given  fractions. 

Rule.  —  Reduce  the  fractions,  if  necessary,  to  their  lowest  terms. 
Tlienfind  the  least  common  multiple  of  the  numerators,  which,  written  over 
the  greatest  common  divisor  of  the  denominators,  will  give  the  least  com- 
mon multiple  required. 

Note.  —  Another  method  is  to  reduce  the  fractions,  if  necessary,  to  their  least 
comvwn  denominator,  and  then  finding  the  least  common  multiple  of  the  numerators, 
and  writing  that  over  the  least  common  denominator. 

ExAjiPLES  FOR  Practice. 

2.  What  is  the  least  common  multiple  of  ^f ,  f ,  and  ^  ? 

Ans.  4|-. 

3.  What  is  the  least  number  that  can  be  exactly  divided  by 
TV,2i,  5,  6^,  and^V?  Ans.  95. 

4.  What  is  the  smallest  sum  of  money  for  which  I  could 
purchase  a  number  of  bushels  of  oats,  at  $  -^^  a  bushel ;  a  num- 
ber of  bushels  of  corn,  at  $  f  a  bushel ;  a  number  of  bushels  of 
rye,  at  $  IJ-  a  bushel ;  or  a  number  of  bushels  of  wheat,  at 
$  2^  a  bushel ;  and  how  many  bushels  of  each  could  I  pur- 
chase for  that  sum  ? 

Ans.  $22^;  72  bushels  of  oats;  36  bushels  of  com;  15 
bushels  of  rye ;  10  bushels  of  wheat. 

5.  There  is  an  island  10  miles  in  circuit,  around  which  A  can 
travel  in  f  of  a  day,  and  B  in  ^  of  a  day.  Supposing  them  each 
to  start  together  from  the  same  point  to  travel  around  it  in  the 
same  direction,  how  long  must  they  travel  before  coming  together 
again  at  the  place  of  departure,  and  how  many  miles  will  each 
have  traveled  ?  Ans.  6^  days ;  A  70  miles  ;  B  60  miles. 

168.  The  rule  for  finding  the  least  common  multiple  of  fractions  ?  Whr 
is  not  the  least  common  multiple  of  the  numerators  the  least  common  mul- 
tiple of  the  fractions  1 


miscella:s^eous  exercises.  169 

miscellaneous  exercises. 

1.  What  are  the  contents  of  a  field  76/^  rods  in  length,  and 
18f  rods  in  breadth  ?  Ans.  8A.  3R.  30^p. 

2.  What  are  the  contents  of  10  boxes  which  are  7f  feet  long, 
1^  feet  wide,  and  1^  feet  in  height? 

Ans.  169^^  cubic  feet. 

3.  From  -/^  of  an  acre  of  land  there  were  sold  20  poles  and 
200  square  feet.     What  quantity  remained  ? 

Ans.  22075ft. 

4.  What  cost  -f^  of  an  acre  at  $  1.75  per  square  rod  ? 

Ans.  $  236.92-^j. 

5.  What  cost  f'g-  of  a  ton  at  $  15f  per  cwt.  ? 

Ans.  $  49.73|f. 

6.  Wliat  is  the  continued  product  of  the  following  numbers : 
14|,  llf,  5|,  and  10^  ?  Ans.  9184. 

7.  From  -fj  of  a  cwt.  of  sugar  there  was  sold  |-  of  it ;  what  is 
the  value  of  the  remainder  at  $  0.12f  per  pound  ? 

Ans.  $  3.18|. 

8.  What  cost  19f  barrels  of  flour  at  $  7f  per  barrel  ? 

Ans.  $  143f . 

9.  Bought  a  piece  of  land  that  was  47y\  rods  in  length,  and 
29/^  in  breadth  ;  and  from  this  land  there  were  sold  to  Abijah 
Atwood  5  square  rods,  and  to  Hazen  Webster  a  piece  that  was  5 
rods  square  ;  how  much  remains  unsold  ? 

Ans.  1366||  square  rods. 

10.  From  a  quarter  of  beef  weighing  175flb.  I  gave  John 
Snow  4  of  it ;  §  of  the  remainder  I  sold  to  John  Cloon.  What 
is  the  value  of  the  remainder  at  8^  cents  per  pound  ? 

Ans.  $'2.04|f. 

11.  Alexander  Green  bought  of  John  Fortune  a  box  of  sugar 
containing  4751b.  for  $  30.  He  sold  ^  of  it  at  8  cents  per  pound, 
and  f  of  the  remainder  at  10  cents  per  pound.  What  is  the 
value  of  what  still  remains  at  12^  cents  per  pound,  and  what 
does  Green  make  on  his  bargain  ? 

.         (  Value  of  what  remains,  $  13.19|-. 
(  Green's  bargain,  $  16.97§-. 

12.  What  cost  -jJ^^j-  of  an  acre  at  $  14f  per  acre  ?    Ans.  $  2. 

13.  Multiply  I  of  T-\  of  ii  ^7  tt  of  U  of  H-       ^^^s.  ^V- 

14.  What  are  the  contents  of  a  board  llf  inches  long,  and  4^ 
inches  wide  ?  Ans.  49-ff  square  inches. 

15 


170  FRACTIONS   OF   DKN'OillXATE  NUMBERS. 

15.  Mary  Brown  had  $  17.87^  ;  half  of  this  sum  was  given  to 
the  missionary  society,  and  f  of  tlie  remainder  she  gave  to  the 
Bible  society  ;  what  sum  has  she  left  ?  Ans.  §  3.57^. 

16.  What  number  shall  be  taken  from  12f,  and  the  remainder 
midtipUed  by  10^,  that  the  product  shall  be  50  ? 

17.  What  number  must  be  multiplied  by  7|,  that  the  product 
may  be  20  ?  Ans.  2||. 

18.  What  are  the  contents  of  a  box  8^2-  feet  long,  3J-^  feet 
wide,  and  2yV  feet  high  ?  Ans.  68|U|-  feet. 

19.  On  f  of  my  field  I  plant  com ;  on  §  of  the  remainder  I 
sow  wheat ;  potatoes  are  planted  on  f  of  what  still  remains ;  and 
I  have  left  two  small  pieces,  one  of  which  is  3  rods  square,  and 
the  other  contains  3  square  rods.     How  large  is  my  field  ? 

Ans.  lA.  OR.  29p. 


REDUCTION   OF   FRACTIONS   OF   DENOMINATE  NUMBERS. 

169.    To  reduce  from  a  higher  to  a  lower  denomination. 

Ex  1.    Reduce  j-xVa  ^^  ^  pound  to  the  fraction  of  a  farthing. 

Ans.  f  far. 

OPEKATION. 

1     X20_^_         ^X12_^  240    X  4  _  960  _4 

l280  ~2160'         2160  ""2160'         2100  ~"2Igo"'~9    "* 

opERATioM  BT  cANCELt^ATioN.  SincB    20s.    make    a    pound, 

1  X  ^0  X  t/i  X  4  there  must  be  20  times  as  many 

"  =  ^far.     shilUncrs  as  pounds  ;   we  therc- 

Xi0  fore  muUiply  ^J^  by  20,  and 

9  obtain  -jfe-g^s.  ;    and  since   12d. 

make  a  shilling,  there  will  be  12  times  as  many  pence  as  shillings; 
hence  we  multiply -j^l^  by  12,  and  obtain  tj-jW^-  Again,  since  4far. 
make  a  penny,  there  will  be  4  times  as  many  farthings  as  pence  ;  we 
therefore  multiply  ^^Yo  ^Y  "*'  ^^^  obtain  •j'^^'^jfar.  =  |^fai-.,  Ans. 

Rule.  —  Multiply  the  given  fraction  by  the  same  numbers  that  would 
be  employed  in  reduction  of  whole  numbers  to  the  lower  denomination 
required. 

169.  The  rule  for  rcdurinp  a  fraction  of  a  higher  denomination  to  the  frac- 
tion of  a  lower  ?  Explain  tlic  <)j)erati()ns  ?  Docs  this  process  differ  in  prin- 
ciple from  reduction  of  whole  (Icnominate  numbers  ? 


REDUCTION.  171 

ExiUiPLEs  FOR  Practice. 

2.  Reduce  x^fVfi  ^^  ^  pound  to  the  fraction  of  a  farthing, 

Ans.  -f|, 

3.  What  part  of  a  penny  is  y*^  of  a  shilling  ?         Ans.  ^|. 

4.  What  part  of  a  grain  is  ^-g^jj  of  a  pound  Troy  ? 

Ans.  f. 

5.  What  part  of  an  ounce  is  x^Vf  ^^  ^  ^^*'-  ^         Ans.  §^. 

6.  Reduce  y^j'^-g-  of  a  furlong  to  the  fraction  of  a  foot. 


Ans.  ^. 


7.  Wliat  part  of  a  square  foot  is  -^s^is  of  an  acre  ? 

Ans.  ^. 

8.  What  part  of  a  second  is  ^^^^  of  a  day  ?         Ans  f  |^. 

9.  What  part  of  a  peck  is  y^y  of  a  bushel  ?  Ans.  ^. 
10.  What  part  of  a  pound  is  g-^^  of  a  cwt.  ?  Ans.  ^. 

17®.   To  reduce  from  a  lower  to  a  higher  denomuiation. 
Ex.  1.  Reduce  f  of  a  farthing  to  the  fraction  of  a  pound. 

Ana 1 


Ans.    2^y^-iy 


OPERATION. 


Tx4~36'     36X12~4^'        432x20~  8C40        ~  2160    ' 

OPERATION  BY  CANCELLATION.  Slncc  4far.  make  a  penny,  there 

^                       1  will   be  ^  as  many  pence  as   far- 

= £.     things;  therefore  we  divide  the  ^ 


9  X  4  X  12  X  20       2160  by  4,  and  obtain    *  d.     And  since 

12d.  make  a  shilling,  there  will  be  ^  as  many  shilhngs  as  pence  ; 
hence  we  divide  A  by  12,  and  obtain  ^f^s.  Again,  since  20s.  make  a 
pound,  there  will  be  -^  as  many  pounds  as  shillings ;  therefore  we 
divide  -^^  by  20,  and  obtain  -g^Vfr  ^  =  ■JTeir  ^  ^^^  *^^  answer. 

Rule.  —  Divide  the  given  fraction  by  the  same  numbers  that  would 
be  employed  in  reduction  of  whole  numbers  to  the  higher  denomina- 
tion required. 

Examples  for  Practice. 
2.  Reduce  ^  of  a  grain  Troy  to  the  fraction  of  a  pound. 

TtrtTFTT- 


Ans.  — -1- 


170.  Do  you  multiply  or  divide  to  reduce  a  fraction  of  a  \o\-rcT  denomina- 
tion to  the  fraction  of  a'higher  1     What  is  the  lul'j  ? 


172  FRACTIONS   OF   DENOMINATE   NUMBERS. 

3.  What  part  of  au  ounce  is  -f^  of  a  scruple  ?        Ans.  ^\j. 

4.  What  part  of  a  ton  is  ^  of  an  ounce  ?  Ans.  itj^^tj. 

5.  What  part  of  a  mile  is  f  of  a  rod  ?  Ans.  -j^^. 

6.  What  part  of  an  acre  is  f  of  a  square  foot  ? 

Ans.  -g-j^j-^. 

7.  What  part  of  a  day  is  f  f  of  a  second  ?        Ans.  sts^tstj- 

8.  What  part  of  3  acres  is  f  of  a  square  foot  ? 

Ans.  2n^rjTiJ- 

9.  What  part  of  3hhd.  is  f  of  a  quart  ?  Ans.  x^^^. 

10.  What  part  of  -^  of  a  sohd  foot  is  a  cube  whose  sides  are 
each  ^  of  a  yard  square  ?  _      Ans.  ^. 

171.    To  find  the  value  of  a  fraction  in  whole  numbers 
of  a  lower  denomination. 

Ex.  1.  What  is  the  value  of  y\  of  1  £  ? 

J  Ans.  7s.  9d.  l^far. 

OPERATION. 

7 
20 

1  8  )  1  4  0  (  7s. 
126 

1  4  Since    1  £  =  20s.,   -^-^   of   a   £   is  ^  of 

1  2  20s.  =  -Ws.  =  7^|s. ;  and  since  Is.  =  12d., 

1  8  TTTTr  9d  H  of  a  sfiUling  is  \i  of  1 2d.  =  Y/d-  =  ^T^d. ; 

16^  =  IJfar.     Therefore  ^£  =  7s.  9d.  IJfar. 

6 
4 


18)24(1  ^far. 
18 

T%  =  i^ar. 

Rule.  —  Multiply  the  numerator  of  the  given  fraction  hy  the  number 
required  to  reduce  it  to  the  next  lotver  denomination,  and  divide  the  pro- 
duct by  the  denominator. 

Then,  if  there  is  a  remainder,  proceed  as  before,  until  it  Li  reduced  to 
the  denomination  required. 

171.  What  is  the  rule  for  finding  tho  value  of  a  fraction  in  whole  numbers 
of  a  lower  denomination  1 


REDUCTION.  173 

Examples  for  Practice. 

2.  What  is  the  value  of  |-  of  a  cwt.  ? 

Ans.  3qr.  21b.  12oz.  7^dr. 

3.  What  is  the  value  of  ^  of  a  yard  ?  Ans.  3qr.  O^na. 

4.  What  is  the  value  of  f  of  an  acre  ? . 

Ans.  IR.  28p.  155ft.  82fln. 

5.  What  is  the  value  of  f  of  a  mile  ? 

Ans.  Ifur.  31rd.  1ft.  lOin. 

6.  What  is  the  value  of  -^^  of  an  ell  EngUsh  ? 

Ans.  Iqr.  l-j^^na. 

7.  What  is  the  value  of  f  of  a  hogshead  of  wine  ? 

Ans.  18gal. 

8.  What  is  the  value  of  -/y  of  a  year  ? 

Ans.  232da.  lOh.  21m.  49^-^860. 

172i  To  reduce  one  denominate  number  to  the  frac- 
tional part  of  another. 

Ex.  1.  What  part  of  l£  is  3s.  6d.  2§far.  Ans.  ■^^£- 

OPERATION.  Since    numbers    compared 

3s.  6d.  2|far.=    512        g   „  must  be  of  the    same  unit, 

oqqa''^  ^^^  reduce  the  3s.  6d.  2§lar. 

1^  =  2880  to    thirds    of   far.,    the    low- 

est denomination  in  the  question,  for  the  numerator  of  the  required 
fraction,  and  l£  to  the  safhe  denomination  for  the  denominator.  We 
then  reduce  this  fraction  to  its  lowest  terms,  and  obtain  ^.£  for  the 
answer. 

Rule.  —  Reduce  the  given  numbers  to  the  loivest  denomination  men- 
tioned in  either  of  them.  Then,  write  the  number  lohich  is  to  become  the 
fractional  part  for  the  numerator,  and  the  other  number  for  the  denomina- 
tor, of  the  required  fraction. 

Note.  —  The  part  that  one  abstract  number  is  of  another  may  be  found  in 
like  manner. 

Examples  for  Practice. 

2.  Reduce  4s.  8d.  to  the  fraction  of  l£.  Ans.  -^-^. 

3.  What  part  of  a  ton  is  4cwt.  3qr.  12lb.  ?  Ans.  ^jjV(j- 

4.  What  part  of  2m.  3fur.  20rd.  is  2fur.  30rd.  ?       Ans.  f  |. 

5.  What  part  of  2A.  2R.  32p.  is  3R.  24p.  ?  Ans.  f 

6.  What  part  of  a  hogshead  of  wine  is  18gal.  2qt.  ? 

Ans.  yW- 

172.  What  is  the  rule  for  reducing  a  denominate  number  to  the  fractional 
part  of  any  other  denominate  number  of  the  same  kind  ? 
15* 


174  FRACTIONS   OF   DENOMINATE  NUMBERS. 

7.  What  part  of  30  days  are  8  days  17h.  20ra.  ?    Ans.  ^f  5. 

8.  From  a  piece  of  cloth  containing  13yd.  Oqr.  2na.  there 
were  taken  5yd.  2qr.  2na.  What  part  of  the  whole  piece  was 
taken  ?  Ans.  f . 

9.  What  part  of  3  yai-ds  square  are  3  square  yards  ? 

Ans.  -J. 

ADDITION  OF  FRACTIONS  OF  DENOMNATE  NUMBERS. 

173i    To  add  fractions  of  denominate  numbers. 

Ex.  1.  Add  f  of  a  pound  to  ^  of  a  shilling, 

Ans.  17s.  lid.  OsVar. 

FiKST  OPERATION.  ^^  fl^^   ^^xe  valuG  of  eacli 

.^1         f  6  -f    1  %         1     ^96  fraction  sepanxtely,  and  add  the 

value  ot  yot   —  1  /         1      2^  j^^   values,   according    to    the 

Value  of  ^s.  =  9      1-J  rule  for  adding  compound  num- 

Y^     ~     ~        bers.     (Art.  101.) 

SECOND   OPEKATION.  -^^  g^.t    ^g^^^^^g 

_         =  ^Z^  £.  the   fraction   of   a 

9  X  20  shilling  to  the  frac- 

?£  +  ^£_  .«-!£_  ,7s.  lid.  OAfar.      J'-   l^^two 
fractions  and  find  the  value  of  their  sum.     (Art.  171.) 

Examples  for  Practice. 

2.  Add  ^j  of  a  pound  to  f  of  a  shiUing. ' 

Ans.  7s.  lid.  3fffar. 

3.  Add  together  \^  of  a  ton,  |-  of  a  ton,  and  t  of  a  cwt. 

Ans.  IT.  14c\vt.  Iqr.  5|^§. 

4.  Add  together  f  of  a  yard,  f  of  a  yard,  -^^  of  a  quarter. 

Ans.  1yd.  2qr.  2na.  Oj^Jin. 

5.  Add  together  y*y  of  a  mile,  f  of  a  mile,  -^^  of  a  furlor.g, 
and  -i-^  of  a  yard.  Ahs.  6fur.  29rd.  3yd.  1ft.  0  J^in. 

6.  Add  together  A-  of  an  acre,  4  of  a  rood,  and  f  of  a  square 
rod.  Ans.  lA.  OR.  3p.  IGOft.  102fin. 

7.  Sold  4  house-lots;  the  first  -^  of  an  acre,  the  second  |  of  an 
acre,  the  tliinl  -^2^  of  an  acre,  and  tlio  fourth  ^  of  an  acre  ;  what 
was  tlie  quantity  of  land  in  the  four  lots  ? 

Ans.  3R.  38p.  455^3 ft- 

173.  What  is  the  first  method  of  addlnp  fractions  of  denominate  numbers  ? 
Wliat  is  tlic  second  ? 


FIRST   OPERATION. 

s.         d. 

far. 

Value  of 

§-£      =17      1 

2&far 

Value  of 

A£    =     7      3 

ItV 

9  10 

m 

SUBTRACTION.  175 

SUBTRACTION   OF  FRACTIONS   OF   DENOMNATE  NUMBERS. 

174.   To  subtract  fractional  parts  of  denominate  numbers. 

Ex.  1.  From  f  of  a  pound  take  -^^  of  a  pound. 

Alls.  9s.  lOd.  Iff  far. 

We  find  the  value  of  each 
fraction  separately,  and  sub- 
tract one  from  the  other,  ac- 
cording to  the  rule  for  sub- 
tracting compound  numbers. 
(Art.  102.) 

SECOND  OPERATION.  We  first  subtract  the  less  fraction  from 

6£ _*  £  =  ^3  £  __  the  greater,  and  then  find  the  value  of 

9s.  lOd.  Iff  far.  *^""'  ^^^'•^^'^-    C^^.  I7i.) 

Examples  for  Practice. 

2.  From  ^  of  a  ton  take  -^j  of  a  cwt. 

Ans.   11  cwt.  Oqr.  T-j-^^b. 

3.  From  ^  of  a  mile  take  y^g-  of  a  furlong. 

Ans.  5fur.  33rd.  5ft.  6iu. 

4.  From  ^  of  an  acre  take  f  of  a  rood. 

Ans.  3R.  IGp.  154ft. 

5.  From  a  hogshead  of  molasses  containing  100  gallons,  y3_ 
of  it  leaked  out ;  f  of  the  remamder  I  kept  for  my  family  ;  what 
quantity  remaiued  for  sale  ?  Ans.  24gal.  Oqt.  If  gpt. 

6.  The  distance  from  Boston  to  Worcester  is  about  41  miles. 
A  sets  out  from  Worcester,  and  travels  -pj-  of  this  distance  to- 
wards Boston ;  B  then  starts  from  Boston  to  meet  A,  and,  having 
travelled  |  of  the  remaining  distance,  it  is  required  to  find  the 
distance  between  A  and  B.  Ans.  12m.  6fur.  9rd.  5ft.  9fiu. 

7.  A  agrees  to  labor  for  B  365  days;  but  he  was  absent  on 
account  of  sickness  4-  part  of  the  time ;  he  was  also  obliged  to  be 
employed' in  his  own  business  y3_  of  the  remaining  time  ;  required 
the  time  lost.  Ans.   137da.  llh.  13m.  14f|sec. 

8.  From  11  acres,  33  poles,  lOl^V  fc^'t  of  land.  T  sold  |  to  A, 
•^  of  the  remainder  to  B,  and  four  house-lots,  each  144  feet  square, 
to  C ;  what  is  the  value  of  the  remainder,  at  8^  cents  ]-»er  square 
foot?  Ans.  $3937.89/^. 

1 74.  What  is  the  first  method  of  subtracting  fractions  of  denominate  num- 
bers i     The  second  ? 


176  QUESTIONS   BY   ANALYSIS. 

QUESTIONS   TO   BE  PERFORMED   BY  ANALYSIS. 

1.  If  one  yard  of  cloth  cost  $  4.40,  what  "will  f  of  a  yard  cost? 

Illustration.  —  If  1  yard  cost  S4.40,  -J^  of  a  yard  -will  cost  \  of 
S  4.40,  or  S  0.88  ;  and  |  will  cost  4  tunes  $  0.88,  or  $  3.52,  Ans. 

2.  If  a  barrel  of  flour  cost  $  7.80,  what  will  -^jj  of  a  barrel 
cost?  Ans.  $  2.34. 

3.  If  a  load  of  hay  cost  $  17.84,  what  will  |  of  a  load  cost  ? 

,  Ans.  $  15.61. 

4.  If  $  786.63  are  paid  for  a  cargo  of  wheat,  what  is  the  cost 
of  1^  of  the  cargo?  Ans.  $  665.61. 

5.  What  is  J-^  of  $  87.50  ?  Ans.  $  80.20|. 

6.  What  is  f  of  17£  I83.  9d.?  Ans.  13£  9s.  Ofd. 

7.  What  is  ^  of  3T.  16cwt.  3qr.  231b.? 

Ans.  2T.  3cwt.  3qr.  23flb. 

8.  What  is  |  of  27A.  3R.  33p.?  Ans.  12A.  IR.  28p. 

9.  Il"  $3.52  are  paid  for  f  of  a  yard  of  cloth,  what  is  the 
price  of  1  yai-d  ?  Ans.  $  4.40. 

Illustration.  —  If  4  of  a  yard  cost  $  3.52,  \  -will  cost  i^  of  $  3.52, 
or  S  0.88  ;  and  f,  or  a  whole  yard,  will  cost  5  times  $  0.88,  or  $4.40, 
Ans. 

10.  If  -fV  of  a  barrel  of  flour  cost  $  2.34,  what  will  be  the  cost 
of  a  whole  barrel  ?  Ans.  $  7.80. 

11.  When  $  15.57 J-  are  paid  for  f  of  a  ton  of  hay,  what  will 
1  ton  cost  ?  Ans.  $  17.80. 

12.  When  -f^  of  a  cargo  of  flour  cost  $  665.50,  what  sum  will 
pay  for  the  whole  cargo  ?  Ans.  $  786.50. 

13.  If  $73.60f  are  paid  for  \}  of  a  ton  of  potash,  what  sum 
must  be  paid  for  a  ton  ?  Ans.  %  80.30. 

14.  Bought  f  of  a  bale  of  broadcloth  for  13£  9s.  0  Jd. ;  Avhat 
would  have  been  the  cost  of  the  whole  bale  ? 

Ans.  17£  18s.  9d. 

15.  If  -j*ij-  of  an  acre  produce  18cwt.  Oqr.  121b.  of  hay,  what 
quantity  will  a  whole  acre  produce?         Ans.  77cwt.  Oqr.  lib. 

16.  Bought  I  of  a  lot  of  land  containing  12A.  IR.  30|p. ; 
what  were  the  contents  of  the  whole  lot  ? 

Ans.  27A.  3R.  39^p. 

17.  If|iJ  of  a  ton  of  potash  cost  $80.20^,  what  is  the  value 
of  a  ton  ?  Ans.  $  87.50. 


QUESTIONS   BY   ANALYSIS.  177 

18.  If  1^  of  a  cwt.  of  sugar  cost  $5.40,  what  is  the  value  of  | 
of  a  cwt.  ? 

Illustkation.  —  If  f  of  a  cwt.  cost  S  5.40,  \  will  cost  |-  of  S  5.40, 
or  S  1.80  ;  and  |,  or  a  cwt,  will  cost  4  times  !$  1.80,  oi-  S  7.20.  Now, 
if  Icwt.  cost  $  7.20,  1  of  a  cwt.  will  cost  J-  of  $7.20,  or  $  0.80  ;  and  | 
will  cost  7  times  $  0.80,  or  $  5.60,  Ans. 

19.  If  -^  of  a  pound  of  ipecacuanha  cost  $2.52,  what  is  tlie 
value  of  1^  of  a  pound  ?  Ans.  $  1.76. 

20.  When  $  80  are  paid  for  ^  of  an  acre  of  land,  what  cost  | 
of  an  acre  ?  Ans.  $  93.3o^. 

21.  If  y^g-  of  a  carding-mill  are  w-orth  $  631.89,  what  are  y^ 
of  it  worth?  Ans.  $401.20. 

22.  If  I"  of  a  ship  and  cargo  are  valued  at  $  141.52,  what  are 
/j  of  them  worth  ?  '  Ans.  $  30.50. 

23.  If  the  value  of  f  of  a  farm  containing  1787^^  acres  is 
$  1728,  what  is  the  price  of  ^  of  the  remainder  ? 

Ans.  $  2304. 

24.  E.  Carter's  garden  is  17^^  ^^^^  ^ong,  and  Hx^ir  I'otls  wide. 
He  disposes  of  f  of  it  for  $  82.80  ;  what  is  the  value  of  f  of  the 
remainder  ?  Ans.  $  41.40. 

25.  Wlien  26£  12s.  6d.  are  paid  for  f  of  a  bale  of  cloth, 
what  sum  should  be  paid  for  ^  of  the  remainder  ? 

Ans.  18£  12s.  9d. 

26.  If  7cwt.  of  sugar  cost  $28.14,  what  will  9|-cwt.  cost? 

Illustration.  —  If  7cwt.  cost  $28.14,  Icwt  will  cost  ^  of  S  28.14, 
or  S  4.02.  In  9-| cwt.  there  are  -^^-cwt. ;  and  if  Icwt.  cost  $  4.02,  icwt. 
will  cost  1  of  $4.02,  or  $0.67,  and  -^  will  cost  59  times  $0.67,  or 
$  39.53,  Ans. 

27.  If  three  tons  of  hav  cost  $  49,  what  will  7^j-  tons  cost  ? 

Ans.  $  120.27xV 

28.  Gave  $78.80  for  11  tons  of  coal;  what  should  I  erive  for 
3|tons?  Ans.  $24.07|J- 

29.  Paid  37£  18s.  lOd.  for  3  bales  of  velvet ;  what  was  the 
cost  of  5f  bales  ?  Ans.  67£  19s.  Q{ii\. 

30.  Gave  $  40  for  5  yards  of  broadcloth ;  what  was  the  prie(3 
of  19^  yards?  Abs.  $156.57f 

31.  Paid  $360  for  20  barrels  of  beer;  what  must  be  given 
for  43|  barrels  ?  Ans.  $  789. 

32.  If  7  bushels  of  rye  cost  $8.75,  what  cost  18/-J-  bushels  ? 

Ans.  S  23.29-i^. 


178  QUESTIONS   BY    ANALYSIS. 

33.  Paid  $19.80  for  3  yards  of  broadcloth;  what  sum  must 
be  given  for  llf  yards  ?  An?.  $  7G.37|. 

34  If  9|cwt.  of  sugar  cost  $39.53,  what  must  be  paid  for 
7cwt.  ? 

Illtjstratiox.  —  In  9|cwt.  there  are  ^cwt.  If  -^wt.  cost  $  39.53, 
^cwt.  will  cost  -^  of  S3y.53,or  S0.67;  and  |,  or  Icwt.,  "will  cost  6 
times  S  0.67,  or  S  4.02;  and  7cwt.  will  cost  7  times  $  4.02,  or  $28.14, 
Ads. 

35.  When  $  18f  are  paid  for  3cwt.  of  sugar,  how  much  may 
be  purchased  for  S  1  ?    How  much  for  $  78  ?     Ans.  12j\^YCwt. 

36.  If  3f  tons  of  potash  cost  $  276.18,  what  will  be  the  value 
of  1  ton  ?     Of  75  tons  ?  Ans.  $  6041.43^. 

37.  If  7-^Y  acres  of  land  -cost  $  875,  what  will  one  acre  cost  ? 
What  will  75  acres  cost?  Ans.  $8912.031^. 

38.  If  4f  tons  of  coal  cost  $  70,  what  will  1  ton  cost  ?  What 
will  86  tons  cost?  Ans,  §1376. 

39.  For  27f  acres  of  land  there  were  paid  $  375  ;  what  cost 
1  acre  ?     What  cost  69  acres  ?  Ans.  $  932.43^9^. 

40.  If  4a  tons  of  hay  cost  $  80.50,  what  costs  1  ton  ?  What 
cost  15  tons  ?  Ans.  $  262.50. 

41.  If  7^cwt  of  sugar  cost  $62.37,  what  will  Icwt.  cost? 
What  cost  19cwt.  ?  Ans.  $  160.93. 

42.  If  7f  yards  of  cloth  cost  $  13.95,  what  will  be  the  value 
of  llf  yards  ? 

Illustration.  —  In  7|  yards  there  are  4^  of  a  yard.  If  4^  of  a 
yard  cost  $  13.95,  -J-  will  cost  -^^  of  S  13.95,  or  S  0.45  ;  and  |,  or  1  yard, 
will  cost  4  times  S  0.45,  or  S  1.80.  In  11 1-  yards  there  are  J-^*  of  a 
yard.  If  1  yard  cost  $  1.80,  ^  of  a  yard  will  cost  -^  of  S  1.80,  or  $  0.20  ; 
and  -L^a  will  cost  103  times  $  0.20,  or  $  20.60,  Ans. 

43.  When  $  668.50  are  paid  for  17y\  acres,  what  would  be 
the  value  of  89|  acres  ?  Ans.  $  3457.30. 

44.  If  $  1738  are  given  for  19^  tons  of  iron,  what  will  be  the 
cost  of  37T*r  tons  ?  Ans.  $  3288. 

45.  Paid  $  llj  for  1128  feet  of  boards  ;  how  many  could  I 
have  purchased  for  $  119^  ?  Ans.  11480  feet. 

46.  For  3|  tons  of  potash  I  received  116cwt.  of  sujrar ;  re- 
quired the  quantity  of  sugar  that  may  be  received  for  11^  tons 
of  potash.  Ans.  376cwt. 

47.  For    11 J   tons  of  potash  I   received  376cwt.  of  sugar; 


MISCELLANEOUS   QUESTIONS.  179 

required  the  quantity  of  sugar  that  should  be  received  for  3| 
tons.  Ans.  llGcwt. 

48.  When  $8  are  paid  for  If  yards  of  broadcloth,  how  much 
must  be  given  for  8f  yards  ?  Ans.  $  49. 

49.  Gave  $  414  for  20^^g-  acres  of  land ;  what  shall  be  given 
for  11|- acres?  Ans.  $236. 

MISCELLANEOUS   QUESTIONS  BY  ANALYSIS. 

1.  Sold  a  small  farm  for  $896.50 ;  what  was  received  for  -^ 
of  it  ?     For  tV  of  it  ?     For  jf  of  it  ?  Ans.   $  815. 

2.  Gave  $  17y\  for  3  barrels  of  flour ;  what  cost  1  barrel  ? 
What  37  barrels  ?  Ans.   $  213.03^V 

3.  Sold  a  house  for  $  3687  ;  what  sum  was  received  for  ^  of 
it?  •  Ans.  $3226.12;!. 

4.  Bought  17-j72-  tons  of  hay  for  $  187f  ;  what  is  the  cost  of  ^ 
of  a  ton?  Ans.  $7.61tVtV 

5.  Bought  a  hogshead  of  molasses  for  $  13|- ;  what  cost  -^  of 
it?     What  cost  i?     What  cost  V?  ^"s.  $30.52^. 

6.  When  $  37y\  are  paid  for  100  gallons  of  molasses,  what 
cost  f  of  a  gallon  ?  Ans.  $  0.2 If  f. 

7.  When  12  cents  are  paid  for  -^  of  a  gallon  of  molasses, 
what  will  48tV  gallons  cost?  Ans.  $  16.01|§. 

8.  If  ^  of  a  barrel  of  flour  cost  $  3f ,  what  will  6f  barrels  cost  ? 

Ans.  $481^^. 

9.  When  $  236  are  paid  for  11|  acres,  what  will  be  paid  for 
20J'jy  acres?  Ans.  $414. 

10.  Paid  in  Liverpool  97f£  for  3  bales  of  cloth  ;  how  many 
bales  should  be  received  for  1073f£  ?  Ans.  33  bales. 

11.  If  6|  barrels  of  flour  cost  $48j^^,  what  will  f  of  a  barrel 
cost?  Ans.  $3.28f. 

12.  If  3f  pounds  of  coffee  cost  34  cents,  what  sum  must  be 
paid  for  74^  pounds?  Ans.  $  6.90 j^j-. 

13.  If  2^  tons  of  hay  cost  $  63,  what  will  be  the  cost  of  1G| 
ton*?  Ans.  $3811|. 

14.  If  a.  piece  of  land  3  rods  square  cost  $  17-j-\,  what  will  be 
the  cost  of  4  square  rods  ?  Ans.  $  7  J^. 

15.  Paid  $31^   for  2f  cwt.  of  iron ;  required  the    sura  to  be 
paid  for  689-j*jCwt.  Ans.  $7680f 

1 6.  For  6|  cords  of  wood  J.  Holt  paid  $  63  ;   what  sum  must 
be  paid  for  1  8  cords  ?  Ans.  $  170.10. 

17.  Gave  $243-^  for  96  barrels  of  tar;  what  quantity  could 
be  purchased  for  $  1000  ?  Ans.  394jf  f  f  barrels. 


180  QUESTIONS   BY  ANALYSIS. 

18  Paid  $7888.30  for  8.3j9^  acres  of  wild  land;  what  sum 
did  I  pay  for  each  acre,  and  what  would  be  the  cost  of  7  acres  ? 

Ans.  $G60.80. 

19.  Gave  132£  12s.  for  7f  tons  of  starch;  what  cost  12| 
ton?  ?  Ans.  224£  5.^. 

20.  For  17^  days'  work  I  paid  $  25.44  ;  what  should  be  paid 
for  89^  days' labor  ?  Ans.  $  128.G4. 

21.  Sold  7yV  bushels  of  apples  for  $7.28;  what  should  I  re- 
ceive for  191  J- "bushels  ?  Ans.  $  19.12. 

22.  Paid  $  4355.52  for  49f  pieces  of  carpeting ;  what  did  37f 
pieces  cost  ?  Ans.  $  3294.72, 

23.  If  ^  of  f  of  the  cost  of  the  Capitol  at  Washington  was 
$  300,000,  what  was  the  whole  cost  ?  Ans.  $  2,000,000.  " 

24.  Purchased  7-j-\  thousand  of  boards  for  $  135.80 ;  what 
must  be  paid  for  19|-  thousand  ?  Ans.  $  359.45. 

25.  My  wood-pile  contains  6  cords  and  76  cubic  feet.  If  I 
dispose  of  ^  of  it,  what  is  the  value  of  the  remainder  at  A^  cents 
per  cubic  foot  ?  Ans.  $23.1 4f|. 

26.  I  have  a  field  30  rods  square,  and  having  sold  18  square 
rods  to  S.  Brown,  and  82  square  rods  to  J.  Smith,  what  part  of 
the  field  remained  unsold  ?  Ans.  f . 

27.  Bought  7T.  12cwt.  3qr.  181b.  of  iron,  and  having  sold 
3T.  18cwt.  Iqr.  201b.,  what  is  the  value  of  f  of  the  remainder 
at  5^  cents  per  lb.  ?  Ans.  $  242.59i. 

28.  Bought  37  tons  of  iron  at  $  68.50  per  ton,  for  f  of  which 
I  paid  in  coffee  at  $  8.50  per  cwt.,  and  for  the  remainder  I  paid 
cash.  Required  the  amount  of  cash  paid,  and  also  the  value  of 
the  coffee. 

Ans.  Cash,  $  633.62^  ;  Value  of  the  coffee,  $  1900.87^. 

29.  A  man,  having  received  a  legacy  of  $  7896,  spent  f  of  it 
in  speculations,  and  the  remainder  he  put  in  the  savings  bank, 
where  it  continued  15  yeai's.  It  was  then  found  that  the  sum 
deposited  had  doubled.     Required  the  sum  in  the  bank. 

Ans.  $3948." 

30.  Bought  a  piece  of  broadcloth  for  $  88,  and  sold  j\  of  it 
to  J.  Smith,  and  -f>j  of  the  remainder  to  O.  LiUce  ;  what  is  the 
value  of  the  part  unsold?  Ans.  $37.49^16^5. 

31.  A  gentleman  gave  ^  of  his  estate  to  his  wife,  f  of  the  re-, 
maindcr  to  his  oldest  son,  and  ^  of  what  then  remained  to  his 
daugliter,  who  received  $  750  ;  required  the  whole  estate. 

Ans.   $12,000. 

32.  From  an  acre  of  land  I  sold  two  house-lots,  each  100  feet 
square  ;  what  is  the  value  of  the  remainder,  at  8  cents  per 
square  foot?  Ans.  $  1884.80. 


DECIMAL   FRACTIONS.  181 


.      DECIMAL    FRACTIONS. 

175.  A  Decimal  Fraction  is  a  fraction  whose  denominator 
is  10,  or  the  product  of  several  lO's. 

Decimal  fractions  are  commonly  expressed  by  writing  the  nu- 
merator only,  with  a  point  (.),  called  the  decimal  point,  before  it, 
care  being  taken  to  put  a  cipher  in  any  decimal  place  not  requir- 
ing a  significant  figure  ;  thus, 


■ffj     may  be  written  .9       and  be  read  9  tenths. 
^Vcr  "  "        -99  "         "      99  hundredths. 

T-^lo  "  "        -099        "        "      99  thousandths, 


By  examining  the  foregoing  fractions,  it  will  be  seen  that,  — 
^Sj  ==  .9  can  occupy  only  one  place  while  it  remains  in  the  form 
of  A  proper  fraction  ;  -^^^^  =  .99,  only  two  places  ;  and  xVJV  =" 
.999,  only  three  places ;  fbi",  if  their  numerators  are  increased 
respectively  by  yV  =  •!,  ihjs  =  -^^i  tttW  =  -001,  each  fraction 
becomes  a  unit  or  whole  number.     Hence, 

Hie  first  figure  or  place  of  any  decimal  on  the  right  of  the  point 
is  tenths,  the  second  hundredths,  the  third  thousandths,  S^c. 

176.  The  denominator  of  -^^  =  .9  is  1  with  one  cipher  an- 
nexed ;  the  denominator  of  -f^-fjj  =  .99  is  1  with  two  ciphers 
annexed  ;  the  denominator  of  y^^^5^0-  =  .999  is  1  with  three  ciphers 
annexed.     Hence, 

The  denominator  of  a  decimal  fraction  is  1  with  as  manij 
ciphers  annexed  as  the  numerator  has  places. 

177.  Decimal  fractions  originate  from  dividing  the  u?iif,  first, 
into  10  equal  parts,  and  then  each  of  these  parts  into  10  other 
equal  parts,  and  so  on  indefinitely.  Thus,  1  -j-  10  =  yiy  =  .1 ; 
rV  -^  10  =  ^l^  =  .01  ;  yi^  --  10  =  ^^\,-^  =  .001.     Hence, 

77te  unit  in  decimcd  fractions  is  divided  into  10, 100, 1000,  ^-c, 
equal  parts. 

175.  What  is  a  decimal  fraction  '  How  are  decimal  fractions  commonly 
expressed  ?  What  is  the  first  figure  or  place  of  any  decimal  ?  The  second  ? 
The  third  ?  &c.  AVhy '?  What  must  he  done  when  a  decimal  place  has  no 
significant  figure  to  fill  it?  —  176.  What  is  tlie  denominator  of  a  decimal 
fraction  ?  — - 177.  How  do  decimal  fractions  originate  ? 
10 


182  DECIMAL   FRACTIONS. 

178.  If  ciphers  are  placed  on  the  left  of  decimal  figures, 
between  them  and  the  decimal  point,  those  figures  change  their 
places,  each  cipher  removing  theni  one  place  to  the  right ;  thus, 
.3  =  f'jy,  but  .03  =  y§^,  and  .003  =  xiAttt*     Hence, 

Every  cipher  placed  on  the  left  of  decimal  fyiires,  between  them 
and  the  decimal  point,  decreases  the  value  represented  by  them  the 
same  as  dividing  by  ten. 

179.  If  ciphers  are  placed  on  the  right  of  decimal  figures, 
their  places  are  not  changed ;  thus,  .3  =  f'g-,  and  .30  =  -^^^ 
=  -^iy  =  .3.     Hence, 

Ciphers  placed  on  the  right  of  decimals  do  not  alter  the  value 
represented  by  them. 

Hence,  decimals  may  be  reduced  to  a  common  denominator,  by 
making  their  decimal  places  equal  by  annexing  ciphers, 

NUMERATION. 

180.  The  relation  of  decimals  to  whole  numbers  and  to  each 
other  may  be  learned  from  the  following 

TABLE. 


4 

o3 

■J3 

m 

1 

en 

a 

■ji 

-4-3 

CO 

3 
O 

ap 

Li 

T3 

CO 

.2 

o? 

a 
o 

O 

•3 
«.- 

o 

or* 

73 

03 

3 

4 

u 

en 

a 

1 

CO 

73 
73 

CO 

fi 
3 

73 

a 

en 
3 
O 

tn 

a 
.2 

.a 
.5 

a 

1 

73 
t3 

CO 

C 

o 

^^ 

C 

c 

O 

a 

a 

-«-> 

*o 

C 

a 

O 

c 

c 

1— « 

c 

3 

i^ 

r^ 

3 

a> 

a 

« 

"3 

« 

OJ 

3 

ri 

aj 

3 

>.-i 

CJ 

3 

r3 

S 

1— 1 

H 

H 

w 

H 

P 

P 

H 

a 

H 

H 

w 

S 

H 

w 

M 

7 

6 

5 

4 

3 

2 

1 

• 

2 

3 

4 

5 

6 

7 

8 

9 

3 

<o 

aT 

cT 

oT 

aT 

aT 

cT 

cT 

aT 

«" 

aT 

aT 

aT 

aT 

aT 

aT 

o 

O 

o 

u 

o 

o 

y 

o 

y 

o 

M 

y 

o 

y 

CJ 

o 

in 

c« 

js 

c3 

R3 

c3 

c3 

CS 

rt 

c3 

c3 

efl 

rt 

cS 

cS 

'p. 

'e^ 

"Ph 

"p- 

'E, 

"Ph 

"Ph 

'Ph 

"p, 

"pi 

"p, 

'p- 

"Pi 

."p. 

'p. 

"pl 

u 

b 

u 

t' 

t< 

u 

b 

Li 

Li 

t^ 

b 

L. 

t- 

'   L. 

L 

u 

o 

o 

o 

o 

o 

o 

O 

S 

o 

O 

o 

o 

O 

O 

o 

o 

u 

u 

u 

^ 

Li 

u 

u 

Li 

Li 

Li 

Li 

Li 

Li 

Li 

Li 

u 

<3i 

o 

(H 

<u 

« 

(U 

a> 

s 

a? 

y 

aj 

O 

« 

a> 

« 

V 

r3 

n 

r3 

T3 

'a 

TJ 

'O 

73 

ri 

'O 

'P 

-TS 

73 

■"P 

ri 

u 

u 

t, 

u 

>-< 

b 

b 

L. 

L. 

1^ 

Li 

t- 

Li 

b 

L 

Li 

O 

o 

o 

o 

O 

o 

o 

o 

O 

o 

o 

o 

o 

o 

O 

O 

^ 

•4.^ 

.S3 

-4-> 

^3 

'T3 

-^ 
n 

4J> 

tn 

-^3 

TS 

J3 

•3 

:2 

M 
-»-> 

t^ 

t£> 

lO 

■* 

CO 

(N 

rH 

(M 

CO 

•<*' 

o 

«> 

b* 

00 

C5 

Whole 

"-r 

Numbers. 

Decimals 

'• 

178.  What  effect  have  ciphers  placed  at  the  left  hand  of  decim.als  ?  Why  1 
—  179.  Wliat  effect  if  [)iii(cd  at  the  riglit  hand ?  Why  ?  — 180.  What  may 
bo  learned  from  tlic  table  1 


NOTATION.  183 

A  Mixed  Number  is  a  whole  number  and  decimal  in  a  single  ex- 
pression. 

The  preceding  table  consists  of  a  whole  number  and  decimal 
■forming  a  mixed  number.  The  part  on  the  left  of  the  decimal 
point  is  the  whole  number,  and  that  on  the  right  the  decimal. 
The  decimal  part  is  numerated  from  the  left  to  the  right,  and  its 
value  is  expressed  in  words  thus:  Two  hundred  thirty-four  mil- 
lions five  hundred  sixty-seven  thousand  eight  hundred  ninety- 
three  billionths.  And  the  mixed  number  tlms :  Seven  milUons 
six  hundred  fifly-four  tliousand  three  hundred  twenty-one,  and 
two  hundred  thirty-four  millions  five  hundred  sixty-seven  thou- 
sand eight  hundi'ed  ninety-three  billionths.     Hence  the 

Rule.  —  Read  the  decimal  as  though  it  were  a  whole  number,  giving 
it  the  name  of  the  right-hand  order. 

Note.  —  A  decimal  with  a  common  fraction  annexed  constitutes  a  complex 
decimal ;  as,  .6^,  read  6^  tenth. 

Write  in  words,  or  read  orally,  the  following  figures  :  — 


1. 

.5 

5. 

.3001 

9. 

.72859 

2. 

.42 

6. 

.0984 

10. 

12.02003 

3. 

.01 

7. 

.00013 

11. 

121.000386 

4. 

.908 

8. 

.82007 
NOTATION. 

12. 

2.3058217 

181  •  Tenths  occupy  the  first  place  at  the  right  of  the  decimal 
point,  hundredths  the  second,  &c.,  and  each  figure  takes  its  value 
by  its  distance  from  the  place  of  units  ;  therefore,  to  write  deci- 
mals, we  have  the  following 

Rule.  —  Write  the  decimal  as  though  it  were  a  whole  number,  supply- 
ing with  ciphers  such  places  as  have  no  signijicant  figures. 

Write  in  figures  the  following  numbers  :  — 

1.  Three  hundred  seven,  and  twenty -five  hundredths. 

2.  Forty-seven,  and  seven  tenths. 

180.  Of  what  does  it  consist?  What  is  the  number  called,  when  taken 
together  ?  What  is  tlie  part  on  the  left  of  the  decimal  point  ?  The  part  on 
the  right?  What  is  the  value  of  the  decimal?  The  value  of  the  mixed 
number  ?  The  rule  for  rcadinjr  decimals  1  —  181.  Upon  what  docs  the  value 
of  a  decimal  figure  depend  ?     The  rule  for  writing  decimals  ? 


184  DECBIAL   FRACTIONS. 

3.  Eighteen,  and  five  hundredths. 

4.  Twenty-nine,  and  three  thousandths. 

5.  Forty-nine  ten  thousandths. 

6.  Eight,  and  eight  milUonths. 

7.  Seventy-five,  and  nine  tenths. 

8.  Two  thousand,  and  two  thousandths. 

9.  Eighteen,  and  eighteen  thousandths. 

10.  Five  hundred  five,  and  one  thousand  six  million ths. 

11.  Three  hundred,  and  forty-two  ten  millionths. 

12.  Twenty -five  hundred,  and  thhty -seven  billionths. 

182.  Decimals,  since  they  increase  from  right  to  left,  and  de- 
crease from  left  to  right,  by  the  scale  of  ten,  as  do  simple  whole 
numbers,  may  be  added,  subtracted,  multiplied,  and  divided,  in 
like  manner. 

ADDITION. 

183.  Ex.  1.  Add  together  5.018,  171.16,  88.133,  1113.6, 
.00456,  and  14.178.  Ans.  1392.09356. 

OPERATION. 

5.0  1  8 

2  7  11  g  We  write  the  numbers  so  that  figures  of  the 

8  81  3  3  same  decimal   place    shall  stand   in   the   same 

^  ^  ^  q'p  column,  and  then,  beginning  at  the  right  hand, 

A  A  /I  -  ^  ^'^^   them   as   whole   numbers,    and   place   the 

.0  0  4  o  b  decimal  point  in  the  result  directly  under  those 

1  4.1  7  8  above. 

1  3  9  2.0  9  3  5  6 

Rule.  —  Write  the  numbers  so  that  Jigures  of  the  same  decimal  place 
shall  gtand  in  the  same  column. 

Add  as  in  whole  numbers,  and  point  off,  in  the  sum,  from  the  ricjht 
hand  as  many  places  for  decimals  as  equal  the  greatest  number  of  deci- 
mal places  in  any  of  the  numbers  added. 

Proof.  —  The  proof  is  the  same  as  in  addition  of  simple  num- 
bers. 

EXAMPI.KS    FOR    PrACTICK. 

2.  Add  together  171.61111,  16.7101,  .00007,  71.0006.  and 
1.167805.  Ans.  260.48it775. 

3.  Add  together  .16711,  1.766,  76111.1,  167.1.  .000007.  nr.d 
1476.1.  Ans.  77756  233117. 

182.  How  do  decimals  increase  and  deciva-so?  How  may  they  be  added, 
Fnhtractcd,  nmltiplicd,  ami  divided  ?  —  18.3.  How  arc  decimals  arranged  for 
addition  1     Tlic  rule  for  addition  of  decimals  f     What  is  the  proof  J 


SUBTRACTION.  185 

4.  Add  together  151.01,  611111.01,  16.'),  6.7,  46.1,  and 
.67896.  Ans.  611331.99896. 

5.  Add  fifty-six  thousand,  and  fourteen  thousandths ;  nine- 
teen, and  nineteen  hundredths  ;  fifty-seven,  and  Ibrty-eight  ten 
thousandths  ;  twenty-three  tliousand  five,  and  four  tenths  ;  and 
fourteen  million  ths..  Ans.  7 9U8 1.6088 14. 

6.  "What  is  the  sura  of  forty-nine,  and  one  hundred  and  five 
ten  thousandths  ;  eighty-nine,  and  one  hundred  seven  thou- 
sandths;  one  hundred  twenty-seven  millionths ;  forty-eight  ten 
thousandths?  Ans.  138.122427. 

7.  What  is  the  sum  of  three,  and  eighteen  ten  thousandths  ; 
one  thousand  five,  and  twenty-three  thousand  forty-three  mil- 
lionths ;  eighty-seven,  and  one  hundred  seven  thousandths ;  forty- 
nine  ten  thousandths ;  forty-seven  thousand,  and  three  hundred 
nine  htmdred  thousandths  ?  Ans.  48095.139833. 

SUBTRACTION. 

184.    Ex.  1.  From  74.806  take  49.054.  Ans.  25.752. 

oPERATiox.  Having  written  the  less  number  tinder  the  greater,  so 

7  4.8  0  6  ^^^^  figures  of  the  same  decimal  place  stand  in  the  same 

4  9  0  5  4  column,  we  subtract  as  in  whole  numbers,  and  place  the 

'- decimal  point  in  the  result,  as  in  addition  of  decimals. 

2  5.7  5  2 

Rule.  —  TFnVe  the  less  number  under  the  greater,  so  that  figures  of  the 
same  decimal  place  shall  stand  in  the  same  column. 

Subtract  as  in  whole  numbers,  and  point  off  the  remainder  as  in  addi- 
tion of  decimals. 

Proof.  —  The  proof  is  the  same  as  in  subtraction  of  simple 
numbers. 

Examples  for  Practice. 


2. 

1  1.0  7  8 
9.8  1 

3. 

4  7.1  17 

8.7  8  1  9  5 

4. 
4  6.1  3 
7.8  9  1  5 

5. 
8  7.1  0  7 
1.1  198  6 

1.2  6  8 

3  8.3  3  5  0  5 

3  8.2  3  8  5 

8  5.9  8  7  1  4 

6.  From  81.35  take  11.678956. 

7.  From  1  take  .876543. 

8.  From  100  take  99.111176. 

9.  From  87.1  take  5.6789. 

Ans.  69.671044. 
Ans.  .123457. 
Ans.  .888824. 
Ans.  81.4211. 

184.  What  is  the  rule  for  subtraction  of  decimals  ?     What  is  the  proof? 
16* 


186  MULTIPLICATION   OF  DECIMALS. 

10.  From  100  take  .001.     '  Ans.  99.999. 

11.  From  seventy-three,  take  seventy- three  thousandths. 

Ans.  72.927. 

12.  From  three  hundred  sixty-five  take  forty-seven  ten  thoU' 
sandths.  Ans.  364.9953. 

13.  From  three   hundred   fifty-seven   thousand   take   twenty- 
eight,  and  four  thousand  nine  ten  millionths. 

Ans.  356971.9995991.. 

14.  From  .875  take  .4.  Ans.  .475. 

15.  From  .3125  take  .125.  Ans.  .1875. 

16.  From  .95  take  .44.  Ans.  .51. 

17.  From  3.7  take  1.8.  Ans.  1.9. 

18.  From  8.125  take  2.6875.  Ans.  5.4375. 

19.  From  9.375  take  1.5.  Ans.  7.875. 

20.  From  .666  take  .041.  Ans.  .625. 

MULTIPLICATION.  ^ 

185.  Ex.  1.  Multiply  18.72  by  7.1.  Ans.  132.912. 

OPEKATioN.  We  multiply  as  in  whole  numbers,  and  point  off  on 

1  8.7  2     the  right  of  the*  product  as  many  figures  tor  decimals 

7  ]^     as  there   are   decimal   figures  iu  the  multiplicand  and 

multiplier. 

18  7  2         The  reason  for  pointing  off  decimals  in  the  product  as 

13  10  4         above  will  be  seen,  if  we  convert  the  multiplicand  and 

multiplier   into   common   fractions,  and   multiply  them 

1  3  2.9  1  2     together.     Thus,  18.72  =  18jT^2_  =  jj^tj  ;  and  7.1  =  1^ 

=  -ri-     Then  ^^^  X  \\  =  ^^\^  =  132/J,^  =  132.912,  Ans.,  tfie 

same  as  iu  the  operation. 

Ex.  2.  Multiply  5.12  by  .012. 

Since  the  number  of  figures  in  the  product  is 
not  equal  to  the  niunbor  of  decimals  in  the  multi- 
y)licand  and  niulti])lier,  we  snojily  the  deficiency 
by  placing  a  cipher  on  the  lefi.  hand. 

The  reason  of  this  process  will  appear,  if  we 
perform  the  question  thus:  .'5.12  r=.5,-y^  =  &^.  and 

•'>12  =  ^1^.  Thonf^^XTH^  =  T^I^  =  -^'^l'»' 
Ans.,  the  same  as  before.  Hence  we  deduce  the 
foUowinpf 

185  In  mulfiplicnfion  of  dcrlmals  how  do  yon  point  off  the  product? 
Tlio  reason  for  it  ?  Wlion  tlio  nimilxT  of  fin-tircf!  in  ilio  pvndiift  is  not  orinal 
to  the  number  of  decimals  in  the  multiiilicand  and  nuiltiiiliov,  what  must  bo 
donol 


OPKRATION. 
5.12 

.0  12 

1024 
5  1  2 

.06144  Ans. 

MULTIPLICATION   OF   DECIMALS.  187 

Rule.  —  Midtlphj  as  in  icltole  numbers,  and  point  off  as  mamj  figures 
for  decimals,  in  the  product,  as  there  are  decimals  in  the  multiplicand  and 
multiplier. 

If  there  he  not  so  many  figures  in  the  product  as  there  are  decimal 
places  in  the  multiplicand  .and  multiplier,  suppbj  the  deficiency  by  prefix- 
ing ciphers. 

Note.  —  To  multiply  a  decimal  by  10,  100,  1000,  &c.,  remove  the  deci- 
mal point  as  many  places  to  the  right  as  there  arc  ciphers  in  the  multiplier ; 
and  if  there  be  not  places  enough  in  the  number,  annex  ciphers.  Thus, 
1.25  X  10  =  12.5;  and  1.7  X  100  =  170. 

Proof.  —  The  proof  is  the  same  as  in  multiplication  of  simple 
numbers. 

Examples  for  Practice. 

3.  Multiply  18.07  by  .007.  Ans.  .12G49. 

4.  Multiply  18.4G  by  1.007.  Ans.  18.58922. 

5.  Multiply  .00076  by  .0015.  Ans.  .00000114. 

6.  Multiply  11.37  by  100.  Ans.  1137. 

7.  Multiply  47.01  by  .047.  Ans.  2.20947. 

8.  Multiply  .0701  by  .0067.  Ans.  .00046967. 

9.  Multiply  47  by  .47.  Ans.  22.09. 

10.  Multiply  eighty-seven  thousandths  by  fifteen  millionths. 

Ans.  .000001305. 

11.  Multiply  one  hundred  seven  thousand,  and  fifteen  teii 
thousandths  by  one  hundred  seven  ten  thousandths. 

Ans.  1144.90001605. 

12.  Multiply  ninety-seven  ten  thousandths  by  four  hundred, 
and  sixty-seven  hundredths.  Ans.  3.886499. 

13.  Multiply  ninety-six  thousandths  by  ninety-six  hundred 
thousandths.  Ans.  .00009216. 

14.  Multiply  one  million  by  one  milhonth.  Ans.  1. 

15.  Multiply  one  hundred  by  fourteen  ten  thousandths. 

Ans.  .14. 

16.  Multiply  one  hundred  one  thousandths  by  ten  thousand 
one  hundred  one  hundred  thousandths.  Ans.  .01020201. 

17.  Multipl)'  one  thousand  fifty,  and  seven  ten  thousandths  by 
three  hundred  five  hundred  thousandths.        Ans.  3.202502135. 

18.  Multiply  two  million  by  seven  tenths.        Ans.  1400000. 

185.  What  is  the  rule  for  multiplication  of  decimals  ?  "What  is  the  proofs 
Hew  do  jou  multiply  a  decimal  by  10,  100,  1000,  &c.  ? 


188  DECIMAL   FRACTIONS. 

19.  Multiply  four  hundred,  and  four  thousandths  by  thirty,  and 
three  hundredths.  Ans.  12012.12012. 

20.  What  cost  461b.  of  tea  at  $  1.125  per  pound  ? 

Ans.  $  51.75. 

21.  What  cost  17.125  tons  of  hay  at  $  18.875  per  ton  ? 

Ans.  $  323.234375. 

22.  Wliat  cost  181b.  of  sugar  at  $  0.125  per  pound  ? 

Ans.  $  2.25. 

23.  What  cost  375.25bu.  of  salt  at  S  0.62  per  busTiel  ? 

Ans.  $  232.655. 

DIVISION. 
186.   Ex.  1.  Divide  45.625  by  12.5.  Ans.  3.65. 

OPERATION.  We  divide  as  in  whole   numbers,  and 

125'i45625r365      since  the  divisor  and  quotient  are  the  two 

'    ''  o  7*  r         ^    *  factors,  which,  being  multiplied  together, 

produce  the  dividend,  we  point  off  two 
decimal  figures  in  the  quotient,  to  make  the 
number  in  the  two  factors  equal  to  the  pro- 
duct or  dividend. 


812 

750 

6  25 
625 

The  reason  for  pointing  off  will  also  be 
seen  by  performing  the  question  with  the 
decimals  in  the  form  of  common  fractions. 
Thus,   45.625   =   453-VA  =   VA¥'   '-^^'i 

-^W  X  T^-V  =  imn  =  ni  =  h'h  =  3.65,  Aus.,  as  before. 
Ex.  2.    Divide   175  by  2.5.  Ans.  .07. 

OPERATION.  "\Ve  divide  as  in  whole  numbers,  and  since  we 

2.5  )  .1  7  5  (  .0  7         have  but  one  figure  in  the  (luotient,  we  jjlace  a 

17  5  cipher  before  it,  which  removes  it  to  the  jiiacc 

of   hundredths,  and   thus    maki*!*  the   decimal 

places    in   the   divisor   and   quotient   equal    to 
those  of  the  dividend. 

Tlic  reason  or  prrfixinrf  the  cipher  will  appear  more  obvious  by 
solving  the  (juestion  witli  the  decimals  in  the  form  of  common  fractions. 
Thus,   .175  =   ,U^,  and   2.5  =2^  =  ff.     Then   ,V/if  -^  f ji  = 

■^f^  .X   M   =  ^V7A  =  lb  =  -0''  A"5.,  as  before.     Hence  the 
following 

186.  In  division  of  di'^imiils  liow  do  you  point  off  the  quotient?  What 
is  the  rea.soii  for  it  1  If  the  di-iinuil  places  of  tlio  divisor  and  quotient  are 
not  equal  to  the  dividend,  what  must  bo  done  1 


DIVISION. 


189 


Rule.  —  Divide  as  in  whole  numbers,  and  point  off  as  many  decimals 
in  the  quotient  as  the  decimals  in  the  dividend  exceed  those  of  the  divisor; 
but  if  there  are  not  as  many,  supply  the  deficiency  by  prefixing  ciphers. 

Note  1.  —  When  the  decimal  places  in  the  divisor  exceed  those  in  the 
dividend,  make  them  equal  by  annexing  ciphers  to  the  dividend,  and  the 
quotient  will  be  a  whole  number. 

Note  2.  —  When  there  is  a  remainder  after  dividing  the  dividend,  ciphers 
may  be  annexed,  and  the  division  continued,  the  ciphers  thus  annexed  being 
regarded  as  decimals  of  the  dividend  ;  to  indicate  in  any  case  that  the  division 
does  not  terminate,  the  sign  plus  (  +  )  can  be  used. 

Note  3.  —  When  a  decimal  number  is  to  be  divided  by  10,  100,  1000,  &c., 
remove  the  decimal  point  as  many  places  to  the  left  as  there  are  ciphers  in 
the  divisor,  and  if  there  be  not  figures  enough  in  the  number,  prefix  ciphers. 
Thus  1.25  -^  10  =  .125  ;  and  1.7  -r-  100  =  .017. 

Proof.  —  The  proof  is  the  same  as  in  division  of  simple 
numbers. 

Examples  for  Practice. 


3.  Divide  183.375  by  489. 

4.  Divide  67.^32  by  32.8. 

5.  Divide  67.56785  by  .035. 

6.  Divide  .567891  by  8.2. 

7.  Divide  .1728  by  10. 

8.  Divide  13.50192  by  1.38. 

9.  Divide  783.5  by  6.25. 

10.  Divide  983  by  6.6. 

11.  Divide  172.8  by  1.2. 

12.  Divide  1728  by  .12. 

13.  Divide  .1728  by  .12. 

14.  Divide  1.728  by  12. 

15.  Divide  17.28  by  1.2. 

16.  Divide  1728  by  .0012. 

17.  Divide  .001728  by  12. 

18.  Divide  116.31  by  1000. 


Ans.  .375. 
Ans.  2.069. 
Ans.  1930.51. 
Ans.  .069255. 
Ans.  .01728. 
Ans.  9.784. 
Ans.  125.36. 
Ans.  148.939+. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 

Ans.  ' 

Ans. 

Ans.  .11631. 

hundred 


19.  Divide    one    hundred    forty-seven,  and    eight 
twenty-eight  thousandths  by  nine,  and  seven  tenths. 

Ans.  15.24. 

20.  Divide  seventy-five,  and  sixteen  hundredths  by  five,  and 
forty-tw^  thousand  eight  hundred  one  hundred  thousandths. 

Ans.  13.846+. 


186.  The  rule  for  division  of  decimals  ? 
Note  3  1    What  is  the  proof] 


What  is   note  11    Note  2? 


190  DECIMAL   FEACTIOXS. 

21.  Divide  six  hundred  seventy-eight  thousand  seven  hundred 
dxty-seven  milhonthrf,  by  three  hundi-ed  twenty-eiglit  thousandths. 

Ans.  2.069+. 
EEDUCTION. 

187.    To  reduce  a  common  fraction  to  a  decimal. 
Ex.  1.    Reduce  f  to  a  decimal.  Ans.  .G25. 

OPERATION.  Since  Tve  cannot  divide  the  nu- 

8  )  5.0  (  6  tenths.  merator,  5,  by  8,  we  reduce  it  to 

4  8  tenths   by  annexing  a  cipher,  and 

then  dividing,  we  obtain  6  tenths 

8)20(2  hundredths.  ^^'^.^  remainder  of  2  tenths.     Re- 

i  g  ducing  this  remainder  to  hundredths 

by  annexing  a  cipher,  and  dividing, 

oxjA/cui  11  "<ve  obtain  2  hundredths  and  a  re- 

8)40(5    thousandths.       painder  of  4  hundredths,  which  be- 

^  0  jijg  reduced  to  thousandths  by  an- 

Ans.  .625.  nexing  a  cipher,  and  then  dividing 

Or  thus :  8  )  5.0  0  0  again,  gives  a  quotient  of  5  thou- 

'*^  sandths.     The  sum  of  the  several 
'^  ^  ^                  ^  quotients,  .625,  is  the  answer. 

To  prove  that  .625  is  equal  to  -|,  we  change  it  to  the  foi-m  of  a  com- 
mon fraction,  by  writing  its  denominator  (Ai-t.  176),  and  reduce  it  to 
its  lowest  terms.     Thus,  -j'^^^q  =  -f,  Ans. 

Rule.  —  Annex  cipheis  to  the  numerator,  and  divide  hy  the  denomi- 
nator. Point  off  in  the  quotient  as  many  decimal  places  as  there  have 
been  ciphers  annexed. 

Examples  for  Practice. 

2.  Reduce  f  to  a  decimal.  Ans.  .75. 

3.  Reduce  |^  to  a  decimah  Ans.  .875. 

4.  What  decimal  fraction  is  equal  to  -^^  ?  Ans.  .4375. 

5.  Reduce  -^j  to  a  decimal.  Ans.  .235294-|- 

6.  Reduce  -j^j-  to  a  decimal.  Ans.  .3G3636-|-» 

7.  Reduce  yV  to  a  decimal.  Ans.  .416666-|-. 

Note.  —  In  rcdiicinp:  a  common  friution  to  a  decimal,  when  the  deiioini- 
nator  cont.iins  other  prime  factors  than  2  and  5,  tliere  cannot  be  an  exact 
division  of  the  numerator  ;  hut,  on  continuins;  the  division,  some  figure  or 
figures  of  the  quotient  will  be  contnmally  repeated. 

A  decimal,  of  which  there  is  a  continual  repetition  of  the  same  figure  or 
figures,  is  called  an  infinite  or  circiiJatiug  decimal.  i 

The  figures  that  repeat  arc  called  repctends.     When  the  rcpetend  is  pre- 

187.  How  do  you  reduce  a  common  fraction  to  a  decimal  ?  How  can  you 
prove  the  answer  correct  ?  The  rule  for  reducing  a  common  fraction  to  a 
decimal  ? 


REDUCTION.  191 

ceded  by  anotlier  dceimal,  the  whole  is  called  a  mixed  repetcnd,  and  the 
part  not  rei)eating  is  called  the  Jinite  part.  To  mark  a  repetcnd,  a  dot  (.) 
is  placed  over  the  first  and  last  of  the  repeating  figures.  Thus,  the  answer 
to  example  sixth,  .36,  is  a  repetend ;  and  the  answer  to  example  seventh, 
.•410,  is  a  mixed  repetcnd,  of  which  the  figure  6  is  the  repetcnd,  and  the 
figures  41  the  finite  part. 

To  change  an  infinite  decimal  to  an  equivalent  common  fraction,  we 
ivrite  the  repetend  for  the  numerator,  and  as  many  nines  as  the  repetend  has 
Jiijures  for  the  denominator.     Thus,  .36  =  |-|  =  i\ ;   and  the  mixed  repe- 

.        ]        ,,%         4lTf  875    5' 

tend,  .416=  jy^  =  9 00  —  1^- 

A  decimal  other  than  a  repetend  is  changed  to  the  form  of  a  common  frac- 
tion, simply /)// «r///?j(7  the  denominutor  under  the  given  numerator.  (Art.  176.) 
Tims,  .75  =  -^^Q  =  I ;  .005  =  y^'Vo'  "^  TiTo- 

8.  Reduce  .875  to  a  common  fraction. 

9.  Change  .4375  to  the  form  of  a  common  fraction. 

10.  Change  .72  to  a  common  fraction.  Ans.  y^-j-. 

11.  Change  .135  to  a  common  fraction.  Ans.  -^j. 

12.  What  common  fraction  is  equivalent  to  .235 G2  ? 

Ana     2  3  1?  2  7 


13.  Change  .093  to  an  equivalent  common  fraction. 


Ans.  y'^-. 


188.    To  reduce  a  denominate   number  to  a  decimal 
of  a  higher  denomination. 

Ex.  1.  Reduce  8s.  6d.  3far.  to  the  decimal  of  a  pound. 

Ans.  .428125. 


4 
12 

20 


OPERATION.  AVe  commence  with  the  3far.,  which  we  re- 

3.0  0  duce  to  hundredths,  by  annexing  two  ciphers ; 

•  and  then,  to  reduce  these  to  the  decimal  of  a 

6.7  5  0  0  penny,  we  divide  by  4far.,  since  there  will  be  \  as 

Q  K  c  i^  K  r\  (\  many  hundredths  of  a  penny  as  of  a  farthing, 

_; and  obtain  .75d.     Annexinor  this  decimal  to  the 


.428125        ^^-1  ^®  divide  by  12d.,  since  there  will  be  ^V  as 
many  shillings  as  pence  ;  and  then  the  8s.  and 
this  quotient  by  20s.,  since  there  will  be  -^^  as  many  pounds  as  shillings, 
and  obtain  .428125£  for  the  answer. 

Rule.  —  Divide  the  lowest  denomination,  annexing  ciphers  if  necessa- 
ry, by  that  number  which  will  reduce  it  to  one  of  the  next  higher  denomi- 
nation. Then  divide  as  before,  and  so  continue  dividing  till  the  decimal 
is  of  the  denomination  required. 

187.  "What  is  an  infinite  decimal?  A  repetend'?  A  mixed  repetend? 
How  is  an  infinite  decimal  changed  to  the  form  of  a  common  fraction  1  — 
188.  The  rule  for  reducing  a  denominate  number  to  a  decimal  of  a  higher 
denomination  ? 


192  REDUCTION   OF   DECIMALS. 

Note.  —  The  given  denominate  number  may  also  be  reduced  to  a  fraction 
of  the  required  denomination  (Art.  170),  and  then  this  fraction  to  a  decimal 
(Art.  187).     Thus,  2s.  6d.  =  -5?^  =  i  =  ■125£. 

Examples  for  Practice. 

2.  Reduce  15s.  6d.  to  the  fraction  of  a  pound.         Ans.  .775. 

3.  Reduce  5cwt.  2qr.  141b.  to  the  decimal  of  a  ton. 

Ans.  282. 

4.  Reduce  3qr.  211b.  to  the  decimal  of  a  cwt.  Ans.  .96. 

5.  Reduce  6fur.  8rd.  to  the  decimal  of  a  mile.         Ans.  .775. 

6.  Reduce  3R.  19p.  167ft.  72in.  to  the  decimal  of  an  acre. 

Ans.  .872595-}-. 

189.  To  find  the  value  of  a  decimal  in  whole  numbers 
of  a  lower  denomination. 

Ex.  1.  "What  is  the  value  of  .9875  of  a  pound  ?     Ans.  19s.  9d. 

OPEKATioN.  There  will  be  20  times  as  many  ten  thousandths  of  a 

9  8  7  5  shilling  as  of  a  pound  ;  therefore,  we  multiply  the  deci- 

n  0  mal,  .0875,  by  20,  and  reduce  the  improper  fraction  to 

a  mixed  number  by  pointing  off '  four  figures  on  the 

1  9.7  5  0  0  right,   which   is    dividing    by   its    denominator,    10000. 

1  2  The  figures  on  the  loft  of  the  point  are  shillings,  and 

those  on  the  right  decimals  of  a  shilling.     The  decimal 

9.0  0  0  0  ,7500  we  multiply  by  12,  and,  pointing  off  as  before, 
obtain  9d.,  which,  taken  with  the  19s.,  gives  19s.  9d.  for  the  answer. 

Rule.  —  MuUiphj  the  decimal  by  that  number  which  will  reduce  it  to  the 
next  lower  denomination,  and  point  off  as  in  midtiplication  of  decimals. 

Then,  multiply  the  decimal  part  of  the  product,  and  point  off  as  before. 
Continue  in  like  manner  till  the  decimal  is  reduced  to  the  denominations 
required. 

The  several   whole   numbers  of  the  successive  products   will  be   the 

answer. 

Examples  for  Practice. 

2.  What  is  the  value  of  .628125  of  a  pound  ? 

Ans.  12s.  6Jd. 

3.  "What  is  the  value  of  .778125  of  a  ton  ? 

Ans.  15cwt.  2qr.  61b.  4oz. 

4.  What  is  the  value  oC  .75  of  aii  ell  English  ? 

Ans.  3qr.  3na. 

5.  What  is  the  value  of  .965625  of  a  mile  ? 

Ans.  7fur.  29rd. 


I 


189.  What  is  the  rule  for  finding  the  value  of  a  decimal  in  whole  numbers 
of  a  lower  denomination  7 


MISCELLA^sEOUS  EXEECISES.  193 

6.  What  is  the  value  of  .94375  of  an  acre  ? 

Ans.  3R.  31p. 

7.  What  is  the  value  of  .815625  of  a  pound  Troy  ? 

Ans.  9oz.  15pwt.  18gr. 

8.  What  is  the  value  of  .5555  of  a  pound  apothecaries'  weight.' 

Ans.  6i  5  3  09  19^|gr. 

MSCELLANEOUS  EXERCISES. 

1.  What  is  the  value  of  15cwt.  3qr.  141b.  of  coffee  at  $9.50 
per  cwt.  ?  Ans.  $  150.95^. 

2.  What  cost  17T.  18cwt.  Iqr.  71b.  of  potash  at  $  53.80  per 
ton  ?  Ans.  $  963.88+. 

3.  What  cost  37A.  3R.  16p.  of  land  at  $  75.16  per  acre? 

Ans.  $2844.80-1-. 

4.  What  cost  15yd.  3qr.  2na.  of  cloth  at  $  3.75  per  yard  ? 

Ans.  $  59.53+. 

5.  What  cost  15f  cords  of  wood  at  $  4.62^  per  cord  ? 

"  Ans.  $71.10+. 

6.  What  cost  the  construction  of  17m.  6fur.  36rd.  of  railroad 
at  $  3765.60  per  mile  ?  Ans.  $  67263.03+. 

7.  What  cost  27hhd.  21gal.  of  temperance  wine  at  $15,37^ 
per  hogshead  ?  Ans.  $  420.24+. 

8.  What  are  the  contents  of  a  pile  of  wood,  18ft.  9in.  long, 
4ft.  6in.  wide,  and  7ft.  3in.  high  ?  Ans.  611ft.  1242in. 

9.  What  are  the  contents  of  a  board  12ft.  6in.  long,  and  2ft. 
9in.  wide  ?  Ans.  34ft.  54in. 

10.  Bought  a  cask  of  vinegar  containing  25gal.  3qt.  Ipt.  at 
$0.37  J-  per  gallon  ;  what  was  the  amount  ?  Ans.  $  9.70+. 

11.  Bought  a  farm  containing  144A.  3R.  30p.  at  $  97.62^  per 
acre ;  what  was  the  cost  of  the  farm  ?  Ans.  $  14149.52+. 

12.  Sold  Joseph  Pearson  3T.  18cwt.  211b.  of  salt  hay  at 
$  9.37^  per  ton.  He  having  paid  me  $  20.25,  what  remains 
due?  Ans.  $16.41+. 

13.  If  |-  of  a  cord  of  wood  cost  $5.50,  what  cost  one  cord? 
What  cost  7f  cords  ?  Ans.  $  48.71+. 

14.  If  4^  yards  of  cloth  cost  $  12f,  what  cost  17^  yards  ? 

Ans.  $  46.18+. 

15.  The  ship  Constantine  cost  $  35000 ;  :|-  of  it  was  sold  to 
Captain  Sampson  for  $  9000  ;  ^  of  the  remainder  to  T.  Lamb  for 
$  9200,  and  the  balance  to  another  person  at  a  profit  of  $  500 ; 
what' was  gained  m  the  sale  of  the  whole  ship?      Ans.  $  1200. 

17 


194  PERCENTAGE. 

PER  CENTAGE. 

190t  Percentage  and  Per  Cent,  are  terms  derived  from  per 
centum,  meaning  by  the  hundred. 

Percentage  is  an  allowance,  at  a  certain  rate,  by  the  hundred. 

The  Rate  per  cent,  is  the  rate  of  allowance  by  the  hundred ;  as  1 
per  cent,,  or  1  hundredth. 

The  Basis  of  percentage  is  the  number  on  which  the  percent- 
age is  reckoned. 

Rates  per  cent,  being  hundredths  may  be  expressed  in  the 
same  manner  as  hundredths  in  decimal  fractions.  Thus,  5  per 
cent  is  written  .05 ;  25  per  cent.,  .25,  &c. 

When  the  rate  per  cent,  is  more  than  100  per  cent,  it  is  ex- 
pressed decimally  as  a  mixed  number.  Thus,  103  per  cent, 
equal  to  \%%  is  written  1.03. 

When  the  rate  per  cent,  is  less  than  1  per  cent.,  or  less  than 
one  hundi'edth,  it  may  be  often  conA^eniently  expressed  as  a 
complex  decimal  (Art.  180,  Note).  Thus,  ^  of  1  per  cent,  may 
be  written  .005,  or  .00^  ;  12^  jjer  cent,,  .122,  or  .12  i,  &c. 

Note.  —  The  sign  "/q  is  often  used,  in  business,  instead  of  the  words 
per  cent. 

Examples. 

Write  decimally  2  per  cent. ;  3  per  cent. ;  5  per  cent. ;  6  per 
cent.;  7  per  cent.;  8  per  cent.;  .10  per  cent.;  12  per  cent.; 
15  per  cent, ;  25  per  cent, ;  30  per  cent.  ;  40  per  cent. ;  50  per 
cent. ;  60  per  cent. ;  75  per  cent. ;  100  per  cent. ;  105  per  cent. : 
1JL5  per  cent. ;  0^  per  cent. ;  8|-  per  cent. ;  201  per  cent. ;  \ 
1  per  cent,  ?  ^  of  1  per  cent, ;  |^  of  1  per  cent, ;  -^^  of  1  [ 
cent. ;  ^  of  1  per  cent. 

191  •    To  find  the  percentage  of  any  quantity. 

Ex.  1.  Bought  a  house  for  %  625,  and  sold  it  at  6  per  cent, 
advance  ;  what  did  I  gain  by  the  sale  ?  Ans.  $  37.50. 

190.  From  what  arc  the  terms  percentafro  and  per  cent,  derived,  and  what 
the  mcaninr;  ?  Define  ])ercentft^o.  How  is  the  rate  written  when  more  than 
100  7  How  when  less  than  1  ?  If  the  per  cent,  is  a  fraction,  or  contains 
a  fraction,  what  is  the  fraction,  if  expressed  decimally  ? 


cent. ; 
of 
per 


PERCENTAGE.  195 

opEKATioN.  Since  the  rate  is  6  per  cent,  or 

Sum,  $  6  2  5  jf^  =  .06,  we  multiply  the  $  625 

Kate  per  cent.,  .0  6  by  the  decimal  expression  .06,  and 

-r.  •  ^  „  _  .  ^  point   off  as   in    multiplication    of 

Per  cent,,  or  gam,       $  3  7.5  0  Jiycimal  fractions. 

Rule.  —  Multiply  the  given  quantity  or  number  by  the  rate  per  cent., 
expressed  as  a  decimal,  and  point  off  the  product  as  iti  muUiplication  of 
decimal  fractions.     (Art.  185.) 

Note.  —  If  the  per  cent,  fcontains  a  fraction '  that  cannot  he  expressed 
in  an  exact  decimal,  or,  if  thus  expressed,  would  require  several  figures,  it  is 
more  convenient  to  multiply  by  it  as  a  complex  decimal. 

Examples  for  Practice. 

2.  What  is  2  per  cent,  of  $  325  ?  Ans.  $  6.50. 

3.  What  is  5  per  cent,  of  %  780  ?  Ans.  $  39.45. 

4.  What  is  6  per  cent,  of  $  856.49  ?  Ans.  $  51.389.* 

5.  What  is  Ih  per  cent,  of  765  tons  ?  Ans.  57.375  tons. 

6.  What  is  Of  per  cent,  of  $  5000  ?  Ans.  %  490. 

7.  What  is  I  per  cent,  of  $  1728  ?  Ans.  $  15.12. 

8.  Wliat  is  4^  per  cent,  of  587  yards  ?  Ans.  26.415  yards. 

9.  I  lost  10  per  cent,  of  $  975 ;  how  much  have  I  remain- 
ing ?  Ans.  %  877.50. 

10.  Sent  to  Liverpool  5000  bushels  of  wheat,  which  cost  me 
$  1.25  per  bushel ;  but  25  per  cent,  of  the  wheat  was  thrown 
overboard  in  a  storm,  and  the  remainder  was  sold  at  $  2  per 
bushel ;  what  was  gained  on  the  wheat?  Ans.  $  1250. 

11.  T.  Page  received  a  legacy  of  $8000;  he  gave  19  per 
cent,  of  it  to  his  wife,  37  per  cent,  of  the  remainder  to  his  sons, 
and  $  2000  to  his  daughters ;  what  sum  had  he  remaining  ? 

Ans.  $  2082.40. 

12.  My  tailor  informs  me  it  will  take  10  square  yards  of 
cloth  to  make  me  a  full  suit  of  clothes.  The  cloth  I  am  about 
to  purchase  is  If  yards  wide,  and  on  sponging  it  Avill  shrink  5 
per  cent,  in  width  and  5  per  cent,  in  length.  How  many  yards 
of  the  above  cloth  must  I  purchase  for  my  "new  suit"  ? 

Ans.  6yd.  Iqr.  l/^y^na. 

13.  A  man  having  $  10000,  lost  15  per  cent,  of  it  in  specula- 
tion; what  sum  had  he  remaining?  Ans.  $8500. 

191.  Explain  the  operation  for  finding  the  percentage  of  any  quantity. 
The  reason  for  the  process  ?     The  rule  ? 


196  SIMPLE   INTEREST. 


SIMPLE    INTEREST. 

m 

192.  Interest  is  an  allowance  made  for  the  use  of  money,  or 
for  value  received. 

The  Rate  per  cent,  is  the  sum  paid  for  the  use  of  $  100,  100 
cents,  100  £,  &c.,  for  any  given  time,  but  usually  for  one  year. 

The  Principal  is  the  sum  on  which  the  interest  is  computed. 

The  Amount  is  the  interest  and  principal  added  together. 

Legal  Lllerest  is  the  rate  per  cent,  established  by  law. 

Usury  is  a  higher  rate  per  cent,  than  is  allowed  by  law. 

The  legal  rate  per  cent,  varies  in  the  different  States  and  in 
different  countries. 

In  Maine,  New  Hampshire,  Vermont,  Massachusetts,  Rhode 
Island,  Connecticut,  New  Jersey,  Pennsylvania,  Delaware,  Mary- 
land, Virginia,  North  Carolina,  Tennessee,  Kentucky,  Ohio, 
Indiana,  Illinois,  Iowa,  Missouri,  Arkansas,  Mississippi,  Flor- 
ida, District  of  Columbia,  and  on  debts  or  judgments  in  favor  of 
the  United  States,  it  is  6  per  cent. 

In  New  York,  Michigan,  Wisconsin,  Minnesota,  Georgia,  and 
South  Carolina,  it  is  7  per  cent. 

In  Alabama  and  Texas,  it  is  8  per  cent. 

In  California,  it  is  10  per  cent. 

In  Louisiana,  it  is  5  per  cent. 

In  Canada,  Nova  Scotia,  and  Ireland,  it  is  6  per  cent. 

In  England  and  France,  it  is  5  per  cent. 

Note. — The  legal  rate,  as  above,  in  some  of  the  States,  is  only  that 
which  the  law  allows,  when  no  particular  rate  is  mentioned.  By  special 
agreement  between  parties,  in  Ohio,  Indiana,  Michigan,  Illinois,  Iowa,  and 
Arkansas,  interest  can  be  taken  as  high  as  10  per  cent. ;  in  Florida  and 
Louisiana,  as  high  as  8  per  cent.;  in  Texas  and  WisTonsin,  as  high  as  12 
per  cent.  ;  and  in  California,  any  per  cent.  In  New  Jersey,  by  a  special 
law,  7  per  cent,  may  be  taken  in  the  city  of  Paterson,  and  in  the  counties 
of  Essex,  Hudson,  and  Bergen. 

193,  To  find  the  interest  of  $  1  at  6  per  cent,  for  any 
given  time. 

Since  the  interest  of  $  1  is  6  cents,  or  y§^  of  the  principal,  for 
1  year,  or  12  months,  for  1  month  it  will  be  -^  of  6  cents,  or  ^ 

192.  What  is  interest  ?  What  is  rate  per  cent.  ?  What  is  the  principal  ? 
What  is  the  amount?  What  is  legal  interest  ?  What  is  usury?  What  is 
the  legal  rate  per  cent,  in  the  ditleront  States  ?  In  Canada,  Nova  Scotia,  and 
Ireland  1     In  England  and  Franco  ? 


I 


SIMPLE  INTEEEST.  197 

cent,  equal  to  5  mills,  or  ^^-^  of  the  principal ;  and  for  2  months, 
twice  5  mills,  or  1  cent,  or  y^g-  of  the  principal. 

Since  the  interest  for  1  month,  or  30  days,  is  5  mills,  the  interest 
for  6  days,  or  i  of  30  days,  will  be  1  mill,  or  xinicr  of  the  principal. 

Since  1  day,  2  days,  &c.,  ai'e  ^,  f ,  &c.,  of  6  days,  the  interest 
for  any  number  of  days  less  than  6  will  be  as  naany  sixths  of  a 
mill,  or  six  thousandths  of  the  principal,  as  there  are  days. 

Also,  since  the  interest  for  2  months  is  1  cent,  or  -j-^-^  of  the 

principal,  for  100  times  2  months,  or  200  months,  or  16  years 

8  mo.,  it  will   be   100  cents,  or   equal   to  the  whole  principal. 

Hence,  the 

TABLE. 

Interest  of  $  1,  at  6  per  cent. 
For  12  mo.  =   1    yr.  is  $0.06,   equal  yfij  of  the  principal. 
«      2  mo.  =   ^   yr.   «      0.01,       "       t^ct        "  " 

«      1  mo.  =  tV  yr.  "     0.005,     "      ^^g.       "  « 

«      6  da.  =   I   mo. «     0.001,     «     Tcjuxy       "  " 

1  aa.  —  Tjij-  v.vuv^,        sts^u 

ALSO, 

For  200  mo.  =16  yr.  8   mo.  is  $  1.00,  equal  the  whole  principal. 

"  100  mo.  =  8  yr.  4   mo.  «  0.50,     "  ^  of  the  " 

«  66|  mo.  =:  5  yr.  6f  mo.  "  0.33^,  «  ^      «  « 

"  50  mo.  =  4  yr.  2    mo.  "  0.25,     «  ^      «  « 

«  40  mo.  =  3  yr.  4   mo.  "  0.20,     "  -i      «  " 

«  33^mo.=  2yr.  9^mo.  «  0.1 6f,  «  J-      «  « 

"  25  mo.  =  2  yr.  1    mo.  "  0.125,  "  ^      "  « 

"  20  mo.  =  1  yr.  8    mo.  "  0.10,     «  J^     «  « 

"  16|mo.  =  lyr.  4|mo.  "  O.OSi    "  ^i^-     «  « 

«  10  mo.  =  f  yr.               «  0.05,     «  ^^     "  " 


u 


6f  mo.  =   I  yr.  «      0.03^,  "     ^V     «  « 

"        5  mo.  =  j%  yr.  "      0.025,  «     ^\y      «  " 

«        4  mo.  =    -[  yr.  «      0.02,     «      -giy     "  « 

Ex.  1.   What  is  the  interest  of  $  1  for  2yr.  7mo.  20da.? 


Ans.  $0,158^ 


FiitsT  OPERATION.  The  interest  for  2  years  will  be 

Interest  for    2y.     =.12  twice  as  much  as  for  1  year,  or  12 

"  "      7mo.  =  .035        cents  ;  and  since  the  interest  for  2 

"  "    20da.   =  .003?      months   is  1   cent,  for  7  months   it  • 

! ?     will  be  3i  cents.     And  as  the  in- 

Ans.    $  0.1  5  8^     terest_  for  6  days  is  1  mill,  for  20 
days  it  will  be  3J  mills.     Adding 
the  several  sums,  we  have  S  0.1 58J  for  the  answer. 


193.  Explain  the  operation. 
17* 


198  SIMPLE  INTEREST. 

SECOND   OPERATION. 

Principal,  $  1.0  0 

^  of  the  prin.,  .12  5     Int.  for  2yr.  Imo. 

■^  of  the  prin,,         .0  3  3^  Int.  for  6mo.  20da. 

9  $  0.1  5  8i  Int.  for  2yr.  7mo.  20da. 

The  time,  2y.  7mo.  20da.,  is  equal  to  2y.  Imo.  -\-  6mo.  20da.  Now, 
since  the  interest  on  any  sum,  at  6  per  cent.,  in  200  months  equals  the 
principal,  for  2y.  Imo.,  or  1  of  200  months,  it  will  equal  \  of  the  prin- 
cipal. We,  therefore,  take  \  of  the  principal,  $  1.00,  equal  12  cents 
and  5  mills,  as  the  interest  lor  2y.  Imo.  The  balance  of  time,  Gmo. 
20da.,  or  6|-  mo.,  being  -^^  of  200  months,  we  take  -^^  of  the  principal, 
equal  3  cents  and  31  mills  as  the  interest  for  Gmo.  20da.  We  add  the 
interest  for  the  parts  of  the  whole  time,  and  obtain,  as  by  first  opera- 
tion, $  0.1581,  as  the  whole  interest. 

"  Rule  1.  —  Reckon  6  cents  for  every  year,  1  cent  for  every  two 
MONTHS,  5  mills  for  the  odd  month,  1  mill  for  every  6  days;  and  for 
any  number  of  days  less  than  six,  as  many  sixths  of  a  mill  as  there  are 
days.     Or, 

Reduce  the  years  and  months  to  months,  and  call  half  the  number  of 
months  cents,  and  one  sixth  the  number  of  days  mills.     Or, 

Rule  2.  —  Take  such  fractional  part  or  parts  of  the  principal  as  the 
number  expressing  the  time  is  of  200  months. 

Examples  for  Practice. 

"  2.  What  is  the  interest  of  $  1  for  ly.  4mo.  6da.  ? 

Ans.  $0,081. 

3.  What  is  the  interest  of  $  1  for  ly.  9mo.  12da.  ? 

Ans.  $  0.107. 

4.  What  is  the  interest  of  $  1  for  3y.  Brno.  19da.? 

An?.  $  0.223^ 

5.  What  is  the  interest  of  $  1  for  2y.  Imo   20da.  ? 

Ans.  $0,128^. 

6.  What  is  the  interest  of  $  1  for  7y.  15da.  ? 

Ans.  $  0.422^. 

7.  What  is  the  interest  of  $  1  for  3mo.  28d.  ? 

,  Ans.  $  0.01 9§. 

8.  What  is  the  interest  of  $  1  for  4y.  2rao.  5da.? 

Ans.  $  0.250^. 

9.  Wliat  is  the  interest  of  $  1  for  4mo.  3da.  ?   Ans.  $  0.020^. 
193.  Explain  the  second  operation.     The  first  rule  ?     The  second  rule? 


Interest, 
Principal, 

648  2 
2778 
1852 

463 

$  2  1  9.9  2  5 
926 

.    SIMPLE  miEEEST.  199 

191.  To  find  the  interest  of  any  sum  at  6  per  cent,  for 
any  given  time. 

Ex.  1.  What  is  the  interest  of  $926  for  3y.  llmo.  15da.? 
What  is  the  amount  ? 

Ans.     Interest,  $  219.925  ;  Amount,  $  1145.925. 

OPERATIOK. 

Principal,  $  9  2  6 

Interest  of  $  1,      .2  3  7^ 

We  find  the  interest  of  $  1  for  the 
given  time  to  be  $0,237^.  (Art.  193.) 
Now,  since  the  interest  of  S  1  is  ^  0.237^, 
the  interest  of  $  1)20  will  be  926  times  as 
much ;  therefore  we  multiply  them  to- 
gether. To  find  the  amount,  wfe  add  the 
principal  to  the  interest. 

Amount,  $  1  1  4  5.9  2  5 

Rule.  —  Find  the  interest  of  $1  for  the  given  time;  then  multiply  the 
j)rincipal  hy  the  number  denoting  this  interest,  and  point  off  as  in  multipli- 
cation of  decimal  fractions.     (Art.  185.)  • 

2b  find  the  amount,  add  the  principal  to  the  interest. 

Note.  —  If  the  interest  of  $  1  contains  a  common  fraction,  the  fraction 
may  be  reduced  to  a  decimal,  if  more  convenient.  The  interest  may  also  be 
multiplied  by  the  number  denodng  the  principal,  when  it  is  preferred. 

Examples  for  Practice. 

2.  What  is  the  interest  of  $  197  for  1  year?     Ans.  $  11.82. 

3.  Wliat  is  the  interest  of  $  1 728  for  3  years  ?  Ans.  $  311 .04. 

4.  What  is  the  interest  of  %  69  for  2  years  ?       Ans.  $  8.28. 

5.  WTiat  is  the  interest  of  $  1728  for  1  year,  6  months? 

Ans.  %  155.52. 

6.  What  is  the  interest  of  $  16.87  for  1  year,  8  months  ? 

Ans.  $  1.687. 

7.  Required  the  interest  of  $  118.15  for  2  years,  6  months. 

Ans.  17.722. 

8.  Required  the  interest  of  $97.16  for  1  year,  5  months. 

Ans.  $  8.258. 

9.  Required  the  interest  of  $789.87  for  }.  year,  11  months. 

Ans.  $  90.835. 

194.  Explain  the  operation  for  finding  the  interest  on  any  sum  of  money 
at  6  per  cent,  for  any  given  time.    The  rule  ?    How  do  you  find  the  amount  ? 


200  SIMPLE  INTEKEST. 

10.  Required  the  amount  of  $978.18  for  2  years,  3  months. 

Ans.  $1110.234, 

11.  Eequired  the  amount  of  $  87.96  for  1  month. 

Ans.  $  88.399. 

12.  Required  the  amount  of  $81.81  for  8  years,  4  months. 

Ans.  $  122.715. 

13.  Required  the  amount  of  $  0.87  for  7  years,  3  months. 

Ans.  $  1.248-. 

14.  What  is  the  interest  of  $  1.71  for  2  years,  2  days  ? 

Ans.  $0,205. 

15.  Required  the  interest  of  $  100  for  8  years,  4  months,  1 
day.  Ans.  $50,016. 

16.  Required  the  interest  of  $  3.05  for  2  months,  and  2  days. 

Ans.  $0,031. 

17.  ^Vhat  in  the  interest  of  $  761.75  for  1  year,  2  months,  18 
days?  Ans.  $55,607. 

18.  What  is  the  interest  of  $1728.19  for  1  year,  5  months, 
10  days?  Ans.  $  149.776. 

19.  What  is*he  interest  of  $88.96  for  1   year,  4  months,  6 
days  ?  Ans.  $  7.205. 

20.  What  is  the  interest  of  $  107.50  for  1  month,  29  days  ? 

Ans.  $  1.057. 

195.   To  find  the  interest  of  any  sum  at  any  rate  per 
cent,  for  any  given  time. 

Ex.  1.   What  is  the  interest  of  $26.25  for  2  years,  4  montlis, 
at  7  per  cent.  ?  Ans.  $  4.2875. 

OPEBATION. 

Principal,  $2  6.2  5 

Interest  of  $  1  at  6  per  cent.,  .1  4         "^c  find  the  interest 

on  the  given  sum  at  6 

10  5  0  0  per  cent.,  and  then  add 

2  6  2  5  to  this  interest  ^  of  it- 

self,  the   part  denoted 

Interest  at  6  per  cent.,  $  3.6  7  5  0  by  the  excess  of  the  rate 

^  of  interest  at  6  per  cent.,  .6125  above  6  per  cent. 

Interest  at  7  per  cent.,  $  4.2  8  7  5 


195.  Explain  the  o])cration  for  finding  the  interest  on  any  sum  at  any  rate 
per  cent 


SIMPLE  INTEREST.  201 

Rule.  —  Find  the  interest  of  the  given  sum  at  6  per  cent.,  and  then 
add  to  this  interest,  or  subtract  from  it,  such  a  part  of  itself  as  the  given 
rate  is  greater  or  less  than  6  per  cent.     Or, 

Take  such  a  part  of  the  interest  at  6  per  cent,  as  the  given  rate  is  of 

6  per  cent. 

Note.  —  J  of  the  interest  at  6  per  cent,  will  be  that  at  1  per  cent. ;  |,  that 
at  5  per  cent. ;  |,  that  at  7  per  cent- ;  twice  the  interest  at  6  per  cent.,  that  at 
12  per  cent.,  etc. 

Examples  for  Practice. 

1.  What  is  the  interest  of  $  144  for  one  year  at  7  per  cent.  ? 

Ans.  $  10.08. 

2.  What  is  the  interest  of  $  850  for  1  year,  7  mouths,  18  days, 
at  7  per  cent.  ?  Ans.  $  97.18. 

3.  What  is  the  interest  of  $8G5.75  for  3  years,  9  months,  24 
days,  at  7  per  cent.  ?  Ans.  $  231.299. 

4.  What  is  the  interest  of  $  960.18  for  1  year,  2  months,  at 

7  per  cent.  ?  Ans.  $  78.414. 

5.  Wliat  is  the  interest  of  $  1728.19  for  3  years,  8  months, 
10  days,  at  7  per  cent.  ?  .4ns.  $  446.929. 

6.  What  is  the  interest  of  $  17.90  for  8  months,  4  days,  at  7 
per  cent.  ?  Ans.  $  0.849. 

7.  What  is  the  interest  of  $1165.50  for  5  years,  3  months, 
9  days,  at  7  per  cent.  ?  Ans.  $  430.36. 

8.  What  is  the  interest  of  $  1237.90  for  1  year,  7  months,  3 
days,  at  7  per  cent.  ?  Ans.  $  137.922. 

9.  What  is  the  interest  of  $156.80  for  3  years  and  3  days, 
at  3  per  cent.  ?  Ans.  $  14.151. 

10.  What  is  the  interest  of  $  579.75  for  1  year,  2  months,  2 
days,  at  5  per  cent  ?  Ans.  $  33.979. 

11.  What  is  the  interest  of  $  7671.09  for  2  years,  8  months, 
5  days,  at  8  per  cent.  ?  Ans.  $  1645.02. 

12.  What  is  the  interest  of  $943.11  for  1  month,  29  days,  at 
9iiercent.?  Ans.  $  13.91. 

13.  What  is  the  interest  of  $  975.06  for  2  years,  7  months,  9 
days,  at  8J-  per  cent.  ?  »  Ans.  $  209.82. 

14.  What  is  the  amount  ^f  $1000  for  3  years,  3  months,  29 
days,  at  5  J-  per  cent.  ?  "  Ans.  $  1183.18. 

15.  What  is  the  interest  of  $  765  for  2  years,  9  months,  at  1 
percent.?  Ans.  $21,037. 

16.  What  is  the  interest  of  $  979.15  for  3  years,  2  months,  4 
days,  at  12^  per  cent.  ?  Ans.  $  388.94. 


202 


SIMPLE   INTEKEST. 


196i    Second  method  of  finding  the  mterest  of  any  sum, 
at  any  rate  per  cent.,  for  any  time. 

Ex.  1.  What  is  the  interest  of  $  26.25  for  3  years,  5  months 
and  15  days,  at  8  per  cent. .?  Ans.  $  7.262. 


OPERATION. 

Principal,  $  2  6.2  5 

Kate  per  cent.,  .0  8 

Interest  for  1  year,  2.1  0  0  0 

3 

Int.  for  3  years,  6.3  0  0  0 

Int.  for  4mo.,  ^  of  ly.,  .7000 

Int.  for  Imo.,  ^  of  4mo.,  .17  5  0 

Int.  lor  loda.,  ^  of  Imo.,  .0875 

Int.  for  3y.  5mo.  15da.,  $7.2  6  2  5 


Having  found  the  inter- 
est for  1  year  and  then  for 
3  years,  the  interest  for  5 
months  is  obtained  by  first 
taking  J  of  1  year's  inter- 
est, for  4  months,  and  then 
\  of  this  last  interest,  for 
1  month. 

And  since  15  days  are  ^ 
of  1  month,  -we  take  i  of  1 
month's  interest  for  the  in- 
terest of  15  days,  and  add 
the  several  sums  for  the 
answer. 


RtJLK.  —  First  find  the  interest  for  one  year  hy  multiplying  the  prin- 
cipal by  the  rate  per  cent. ;  and  for  two  or  moi-e  years  multiply  this 
product  by  the  number  of  years. 

Find  the  interest  for  months  by  taking  the  most  convenient  fractional 
part  or  parts  of  one  year's  interest. 

Find  the  interest  for  days  by  taking  the  most  convenient  fractional  part 
or  parts  o/one  month's  interest. 

Note.  —  Many  practical  men  prefer  this  method  of  casting  interest  to 
any  other,  but  in  most  questions  it  is  not  so  expeditious  as  the  preceding. 
The  pupil  may  be  required  to  solve  questions  by  both  methods. 

Examples  fob  Practice. 

2.  What  is  the  mterest  of  $  1775  for  7  years? 

Ans.  $  745.50. 

3.  Wliat  is  the  interest  of  $  987  for  3  years,  6  months  ? 

Ans.  $  207.27. 

4.  Required  the  interest  of  $  69.17  for  4  years,  9  months. 

Ans.  $19,713. 

5.  Required  the  interest  of  $  96.87  for  10  yeai-s,  7  months,  15 
days.  Ans.  $  61.754. 


196.  Explain  the  operation  for  finding  the  interest  of  any  sum,  at  any  rate 
per  cent.,  for  any  time.    What  is  the  mle  1 


SIMPLE  INTEREST.  203 

C).  Required  the  interest  of  $  1.95  for  15  years,  11  months, 
20  days.  Ans.  $  1.868. 

7.  Required,  the  interest  of  $  1789  for  20  years,  1  month, 
25  days.  Ans.  $  2163.199. 

8.  Required  the  interest  of  $  666.66  for  6  years,  10  months, 
13  days.  Ans.  $  274.775. 

9.  What  is  the  amount  of  $  98.50  for  5  years,  8  months  ? 

Ans.  $131.99. 

10.  What  is  the  amount  of  $  168.13  for  8  years,  5  months,  3 
days?  Ans.  $253,119. 

11.  Wliat  is  the  amount  of  $75.75  for  4  years,  2  months,  27 
days  ?  Ans.  $  95.028. 

12.  Required  the  amount  of  $  675.50  for  30  years,  3  months, 
23  days.  Ans.  $  1904.121. 

197.  To  find  the  interest  on  pounds,  shillings,  pence, 
and  farthings,  at  any  rate  per  cent.,  for  any  time. 

Ex.  1.  What  is  the  interest  of  25£  2s.  6d.  for  2  years,  6 
months,  at  6  per  cent.  ?  Ans.  oM  15s.  5d.  2far. 

OPERATION.  We  reduce  the  2s.  6d.  to  the 

25£  2s.  6d.  =  2  5.1  2  5  £  decimal  of  a  pound  (Art.  188), 

Interest  of  1£            .1  5  and,  annexing  it  to  the  pounds, 

multiply  this  principal  by  the  in- 

12  5  6  2  5  terest  of  l£  for  the  given  time. 

2  5  12  5  The  product  is  in  pounds  and  the 

decimal   of  a  pound,  which  we 

6.7  0  o  7  5  X  =  reduce  to  shillings,  pence,  and 

3£  15s.  4d.  2far.  farthings.    (Ai-t.  189.) 

Rule.  —  Reduce  the  shillings,  pence,  and  farthings  to  the  decimal  of  a 
pound,  and  annex  it  to  the  pounds;  then  proceed  as  #1  United  States 
money,  and  reduce  the  decimal  in  the  result  to  a  compound  number. 

Examples  for  Practice. 

2.  What  is  the  interest  of  26£  10s.  for  2  years,  4  months, 
at  5  per  cent.  ?  Ans.  3£  Is.  lOd. 

3.  What  is  the  interest  of  42£  18s.  for  1  year,  9  months,  25 
days,  at  6  per  cent.?  Ans.  4£  133.  7fd. 

4.  What  is  the  interest  of  94£  12s.  6d.  for  4  years,  6  months, 
7  days,  at  8  per  cent.  ?  Ans.  34£  4s.  2f . 

»  • 

197.  How  do  you  find  the  interest  on  pounds,  shillings,  pence,  and  far- 
tiiings  ■?     Repeat  the  rule. 


I 


204  SIMPLE  INTEREST. 

MISCELLANEOUS  EXERCISES. 

1.  What  is  the  interest  of  $  172.50  from  Sept.  25,  1850,  to 
July  9,  1852  ?  Ans.  $  18.515. 

2.  What  is  the  interest  of  $  169.75  from  Dec.  10,  1848,  to 
May  5,  1851?  Ans.  $24,472. 

3.  What  is  the  interest  of  $17.18  from  July  29,  1847,  to 
Sept.  1,  1851  ?  Ans.  $  4.214. 

4.  What  is  the  interest  of  $  67.07  from  April  7,  1849,  to 
Dec.  11,  1851?  Ans.  $10,775. 

5.  Required  the  interest  of  $  117.75  from  Jan.  7,  1849,  to 
Dec.  19,  1851.  Ans.  $  20.841. 

6.  Required  the  interest  of  $847.15  from  Oct.  9,  1849,  to 
Jan.  11,  1853.  Ans.  $  165.476. 

7.  Required  the  interest  of  $7.18  from  March  1,  1851,  to 
Feb.  11,  1852.  Ans.  $0,406. 

8.  What  is  the  interest  of  $976.18  from  May  29,  1852,  to 
Nov.  25,  1855  ?  Ans.  $  204.347. 

9.  I  have  John  Smith's  note  for  $  144,  dated  July  25,  1849  ; 
what  is  due  Mafch  9,  1852  ?  Ans.  $  166.656. 

10.  George  Cogswell  has  two  notes  against  J.  Doe ;  the  first 
is  for  $  375.83,  and  is  dated  Jan.  19,  1850  ;  the  other  is  for 
$  76.19,  dated  April  23,  1851 ;  what  is  the  amount  of  both  notes 
Jan.  1,1852?  Ans.  $499,141. 

11.  What  is  the  interest  of  $  68.19,  at  7  per  cent.,  from  June 
5,  1850,  to  June  11,  1851  ?  Ans.  $  4.852. 

12.  Required  the  amount  of  $79.15  from  Feb.  17,  1849,  to 
Dec.  30,  1852,  at  7^  per  cent.  Ans.  $  102.119. 

13.  What  is  the  amount  of  $89.96  from  June  19,  1850,  to 
Dec.  9,  1851^  at  8^  per  cent.  ?  Ans.  $  100.886. 

14.  A.  Atwood  has  J.  Smith's  note  for  $325,  dated  June  5, 
1849 ;  what  is  due,  at  74-  per  cent.,  July  4,  1851  ? 

Ans.  $  374.022. 

15.  J.  Ayer  has  D.  How's  note  for  $  1728,  dated  Dec.  29, 
1849  ;  what  is  the  amount  Oct.  9,  1852,  at  9  per  cent.  ? 

Ans.  $2160. 

16.  What  is  the  interest  of  $976.18  from  Jan.  29,  1851,  to 
July  4,  1852,  at  12  per  cent.  ?  Ans.  $  167.577. 

17.  What  is  the  amount  of  $175.08  from  May  7,  1851,  to 
Sept.  25,  1853,  at  7  per  cent.  ?  Ans.  $  204.289. 

18.  What  is  the  amount  of  $160  from  Doc.  11,  1853,  to 
Sept.  9,  1854,  at  7  per  cent.  ?  Ans.  $  168.337. 


PARTIAL  PAYMENTS.  205 

PARTIAL  PAYMENTS. 

198.  A  Promissory  Note,  or  note  of  hand,  is  an  engagement,  in 
Writing,  to  pay  a  specified  sum,  eitlier  to  a  person  named  in  tlie 
note,  or  to  his  order,  or  to  the  bearer. 

A  Joint  Note  is  one  signed  by  two  or  more  persons,  who  to- 
gether are  holden  for  its  payment. 

A  Joint  and  Several  Note  is  one  signed  by  two  or  more  persons, 
who  separately  and  together  are  holden  for  its  payment. 

A  Negotiable  Note  is  one  so  made  that  it  can  be  sold  or  trans- 
ferred Ironi  one  person  to  another. 

The  Maker  or  Drawer  of  a  note  is  the  person  who  signs  it. 

The  Payee,  Promisee,  or  Iloldcr  is  the  person  to  whom  it  is  to 
be  paid. 

The  Indorser  of  a  note  is  the  person  who  writes  his  name  upon 
its  back  to  transfer  it,  or  as  guaranty  of  its  payment. 

The  Face  of  a  note  is  the  sum  for  which  it  is  given. 

Partial  Payments  or  part  payments  of  a  note  or  other  obligation, 
being  receipted  for  by  an  entry  on  the  back  of  the  obligation,  are 
called  Indorsements. 

Merchants'  Rule. 

199.  Wlien  settlement  is  made  within  one  year,  merchants 
usually  compute  by  the  following 

Rule.  —  Find  the  amount  of  the  principal  from  the  time  it  "became  due 
until  the  time  of  payment.  Then  find  the  amount  of  each  indorsement 
from  the  time  it  was  paid  until  settlement,  and  subtract  their  sum  from 
the  amount  of  the  principal. 

Note.  —  This  is  the  common  rule  ia  Vermont  for  any  time. 

Ex.  1.     $  1234.  Boston,  Jan.  1,  1853. 

For  value  received,  I  promise  to  pay  John  Smith,  or  order,  on 
demand,  one  thousand  two  hundred  thirty-four  dollars,  with 
interest.  John  T.  Jones. 

Indorsements:  —  March  1,  1853,  received  ninety-eight  dollars. 
June  7,  1853,  received  five  hundred  dollars.  Sept.  25,  1853,  received 
two  hundred  ninety  dollars.  Dec.  8,  1853,  received  one  hundred 
dollars. 

What  remains  due  at  the  time  of  payment,  Jan.  1,  1854? 

Ans.  $  293.12. 

198.  What  is  a  note  ?  A  negotiable  note?  A  joint  note  ?  Who  is  the 
mnker  of  a  note?  Who  tlie  payee  1  Who  the  indorser?  What  are  par- 
tial  payments?  — 199.  What  is  the  rule  for  computing  the  interest  when 
there  arc  partial  payments,  and  settlement  is  made  within  one  year  1 

lb 


206  SMPLE   INTEREST. 

t 

OPERATIOW. 

Principal, S  1234.00 

lat.  from  Jan.  1,  1853,  to  Jan.  1,  1854  (ly.),    .         .         .  74.04 

Amount,        ........ 

First  payment,  March  1,  1853,  .... 

Int.  from  March  1,  1853,  to  Jan.  1,  1854  (lOmo.),  . 
Second  payment,  June  7, 1853,         .... 

Int.  from  June  7, 1853,  to  Jan.  1,  1854  (6mo.  24da.), 
Third  payment,  Sept.  25,  1853,         .... 

Int.  from' Sept.  25,  1853,  to  Jan.  1,  1854  (3mo.  6da.), 
Fourth  payment,  Dec.  8,  1 853,  .        \         .         . 

Int.  from  Dec.  8,  1853,  to  Jan.  1,  1854  (23da.),     . 

Amount  of  payments  to  be  deducted, 

Balance  remains  due  Jan.  1,  1854,       .        .         .        .         .    $233.12 


•                  • 

1308.04 

$98.00 

4.90 

500.00 

17.00 

290.00 

4.64 

100.00 

.38 

• 

$1014.92 

2.     $  987.75.  Trenton,  Jan.  11,  1852. 

For  value  received,  we  jointly  and  severally  promise  to  pay 
James  Dayton,  or  order,  on  demand,  two  months  from  date,  nine 
hundred  eighty-seven  dollars  seventy-Jive  cents,  with  interest  after 
ttoo  months.  John  T.  Johnson. 

Attest,  Isaiah  Webster.  Samuel  Jones. 

Indorsements  :  —  May  1, 1 852,  received  three  hundred  dollars.  June 
6,  1852,  received  four  hundred  dollars.  Sept.  25,  1852,  received  one 
hundred  and  fifty  dollars. 

What  is  due  Dec.  13,  1852  ?  Ans.  $  156.94. 


3.     $  800.  Indianapolis,  July  4,  1852, 

I^or  valiie  received,  I  promise  to  pay  Leonard  Johnson,  or  order, 
on  demand,  eight  hundred  dollars,  with  interest. 

Attest,  Charles  True.  Samuel  Neverpay. 

Indorsements:  —  Aug.  10,  1852,  received  one  hundred  forty-four 
dollars.  Nov.  1,  1852,  received  ninety  dollars.  Jan.  1,  1853,  re- 
ceived four  hundred  dollars.  March  4,  1853,  received  one  hundred 
dollars. 

"What  remains  due  June  1,  1853  ?  Ans.  $  88.02. 

United  States  Rule. 

200t  The  United  States  courts,  and  most  of  the  courts  of  the 
several  States,  adopt  the  following 

199.  How  do  you  explain  the  operation  ? 


PARTIAL  PAYMENTS.  207 

Rule.  —  Compute  the  interest  on  the  principal  to  the  time  when  the 
first  payment  was  made,  which  equals,  or  exceeds,  either  alone  or  with  pre- 
ceding payments,  the  interest  then  due. 

Add  that  interest  to  the  principal,  and  from  the  amount  subtract  the 
payment  or  payments  thus  far  made. 

The  remainder  will  form  a  new  principal ;  on  which  compute  the  in^ 
terest,  proceeding  as  before. 

Note  1.  —  This  rule  is  on  the  principle,  that  neither  interest  nor  pay- 
ment should  draw  interest. 

Note  2.  —  In  New  Hampshire  the  State  Courts  allow  interest  on  "an- 
nual "  interest,  in  the  nature  of  damages  for  its  detention,  from  the  time  it 
becomes  due  till  paid. 


Ex.  1.     $365.50.  Wilmington,  Jan.  1,  1852. 

For  value  received,  I  promise  to  pay  to  John  Dow,  or  order,  on 
demand,  three  hundred  sixty-jive  dollars  fifty  cents,  with  interest. 
Attest,  Samuel  Webster.  John  Smith. 

Indorsements  :  —  June  10, 1852,  received  fifty  dollars.  Dec.  8,  1852, 
received  thirty  dollars.  Sept.  25,  1853,  received  sixty  dollars.  July 
4,  1854,  received  ninety  dollars.  Aug.  1,  1855,  received  ten  dollars. 
Dec.  2,  1855,  received  one  hundred  dollars. 

What  remains  due  Jan.  7,  1857  ?  Ans.  $  92.53. 

OPERATION. 

Principal   carrying   interest  from  Jan.  1,  1852,  to  June  10, 

1852, $365.50 

Interest  from  Jan.  1, 1852,  to  June  10,  1852  (5mo.  9da.),  9.68 

Amount, 375.18 

First  payment,  June  10,  1852, 50.00 

Balance  for  new  principal, 325.18 

Interest  from  June  10,  1852,  to  Dec.  8,  1852  (5mo.  28da.),  9.64 

Amount, 334.82 

Second  payment,  Dec.  8,  1852,         .         .         .        .         .  30.00 

Balance  for  new  principal, 304.82 

Int.  for  Dec.  8,  1852,  to  Sept.  25,  1853  (9mo.  17  days),    .  14.58 

Amount,        .  • 319.40 

Third  payment,  Sept  25,  1853, 60.00 

Balance  for  new  prmcipal, 259.40 

Interest  from  Sept.  25,  1853,  to  July  4,  1854  (9mo.  9  days),  12.06 

Amount, 271.46 

200.  What  is  the  rule  generally  adopted  by  the  several  States  for  comput- 
ing the  interest  on  notes  and  bonds,  when  partial  payments  have  been 
made. 


208  SIMPLE  INTEREST. 

Amount  brouglit  up,    S  271.46 
Fourth  payment,  July  4,  1854, 90.00 

Balance  for  new  principal, 181.46 

Interest  from  July  4,  1854,  to  Aug.  1, 1855  (12mo.  27  days),  11.70 

Interest  from  Aug.  1,  1855,  to  Dec.  2,  1855  (4mo.  1  day),    .  3.66 

Amount,      ..........        196.82 

Fifth  payment,  Aug.  1,  1855  (a  sum  less  than  the 

interest), $10.00 

Sixth  payment,  Dec.  2,  1855  (a  sum  greater  than 

the  interest,    .......  100.00 

110.00 


Balance  for  new  principal,      ......  86.82 

Interest  from  Dec-  2,  1855,  to  Jan.  7,  1857  (13mo.  5  days),  5.71 

Remainsdue  Jan.  7,  1857,        .        .        .        .        .        $92.53 


2.  $  1666.  Philadelphia,  June  5,  1848. 

For  value  received,  I  promise  to  pay  J.  B.  Lippincott  Sf  Co.,  or 
order,  on  demand,  without  defalcation,  one  thousand  six  hundred 
sixty-six  dollars,  with  interest.  John  J.  Shellenherger. 

Attest,  T.  Webster. 

Indorsements :  —  July  4,  1849,  received  one  hundred  dollars.  Jan.  1. 
1850,  received  ten  dollars.  July  4,  1850,  received  fifteen  doUare. 
Jan.  1, 1851,  received  five  hundred  dollars.  Feb.  7,  1852,  received  six 
hundred  and  fifty-six  dollars. 

What  is  due  Jan.  1,  1853  ?  Ans.  $  767.08. 

3.  $9607  Detroit,  Oct.  23,  1850. 

On  demand,  I  promise  to  pay  to  S.  S.  St.  John,  or  order,  nine 
hundred  sixty  dollars,  for  value  received,  with  interest  at  seven 
per  cent.  J^^^'^  Q-  Smith. 

Attest,  H.  F.  Wilcox. 

Indorsements:  — Sept.  25,  1851,  received  one  hundred  forty  dol- 
lars. July  7,  1852,  received  eighty  dollars.  Dec.  9,  1852,  received 
seventy  dollars.     Nov.  8,  1853,  received  one  hundred  dollars. 

What  is  due  Oct.  23,  1854  ?  Aus.  %  807.76. 

4.  $  1000.  New  Toric,  Jan.  1,  1849. 
Two    months  after  date  I  promise  to  pay  to  S.  Durand,  or 


200.  Explain  the  operation. 


PARTIAL  PAYMENTS.  209 

order,  one  thousand  doUars,  for  value  received,  with  interest  after, 
at  seven  per  cent.  Paul  Sampson,  Jr. 

Indorsements:  —  March  1,  1850,  received  one  hundred  dollars. 
Sept.  25,  1851,  received  two  hundred  dollars.  Oct.  9,  1852,  received 
one  hundred  fifty  dollars.  July  4,  1853,  received  twenty  dollars. 
Oct.  9,  1853,  received  three  hundred  dollars. 

What  is  due  Dec.  1,  1854  ?  Ans.  $  567.49. 

Connecticut   Rule. 

201.  The  rule  established  by  the  Supreme  Court  of  the  State 
of  Connecticut. 

Compute  the  interest  to  the  time  of  the  first  payment ;  if  that  he  one  year 
or  more  from  the  time  the  interest  commenced,  add  it  to  the  principal,  and 
deduct  the  payment  from  the  sum  total.  If  there  he  after  payments  made, 
compute  the  interest  on  the  balance  due  to  the  next  payment,  and  then 
deduct  the  payment  as  above ;  and  in  like  manner  from  one  payment  to 
another,  till  all  the  payments  are  absorbed  ;  provided  the  time  between  one 
payment  and  another  be  one  year  or  more. 

But  if  any  payments  be  made  before  one  year's  interest  hath  accrued, 
then  compute  the  interest  on  the  principal  sum  due  on  the  obligation  for 
one  year,*  add  it  to  the  principal,  and  compute  the  interest  on  the  sum 
paid  from  the  time  it  tvas  paid  up  to  the  end  of  the  year ;  add  it  to  the  sum 
paid,  and  deduct  that  sum  from  the  principal  and  interest  added  together. 

If  any  payments  he  made  of  a  less  sum  than  the  interest  arisen  at  the 
time  of  such  payment,  no  interest  is  to  be  computed,  but  only  on  the  prin- 
cipal sum  for  any  period. 


I 


Ex.  1.     $500.  Hartford,  July  1,  1854. 

For  value  received,  I  promise  to  pay  J.  Dow,  or  order,  on 
demand,  five  hundred  dollars,  with  interest.  D.  P.  Page. 

Indorsements :  —  Sept.  1, 1855,  received  one  hundred  dollars.  April 
1, 1856,  received  one  hundred  forty-four  dollars.  Jan.  1,  1857,  received 
ninety  dollars,  fifty  cents.  Dec.  1,  1858,  received  one  hundred  sixty- 
eight  dollars,  five  cents. 

What  is  due  Oct.  1,  1859  ?  Ans.  $  92.40. 

*  If  a  year  extends  beyond  the  time  when  the  note  becomes  due,  find  the 
amount  of  the  remaining  principal  to  the  time  of  settlement ;  find  also  tlie 
amount  of  the  indorsement  or  indorsements,  if  an)',  from  the  time  they  were 
paid  to  the  time  of  settlement,  and  subtract  their  sum  from  the  amount  of 
the  principal. 

201 .  What  is  the  Connecticut  rule  1 
18* 


210  SIMPLE  INTEREST. 

PROBLEMS  IN  LNTEREST. 

202.  A  Problem  is  a  question  proposed  for  solution. 

203.  In  the  preceding  questions  in  interest,  five  terms  or 
tilings  have  been  mentioned  ;  namely,  the  Interest,  Amount,  Rate 
per  cent..  Time,  and  Principal. 

The  investigation  of  these  involves  five  problems :  I.  To  find 
the  interest ;  II.  To  find  the  amount ;  III.  To  find  the  rate  per 
cent. ;  IV.  To  find  the  time ;  V.  To  find  the  principal. 

With  one  exception,  any  three  of  the  preceding  terms  being 
given,  a  fourth  may  be  found  by  the  rules  deduced  from  the  solu- 
tion of  the  problems. 

If,  however,  the  rate  per  cent.,  time,  and  amount  are  given, 
an  additional  rule  is  necessary  to  find  the  principal,  which  will 
form  problem  VI.  ;  but,  from  its  connection  Avith  Discount  (Ai-t. 
210),  its  solution  will  be  deferred. 

The  Problems  I.  and  11.  have  already  been  examined  (Ai't. 
194). 

204.  Problem  III.  To  find  the  rate  per  cent.,  the 
principal,  interest,  and  time  behig  given. 

Ex.  1.  The  mterest  of  %  300  for  2  years  is  $  48 ;  what  is  the 
rate  per  cent.  ?  Ans.  8  per  cent. 

OPERATION.  We  find  the  interest  on  the  prln- 

$  3  0  0  cipal  for  2  years  at  1  per  cent,  and 

,0  2  divide  the  given  interest  by  it. 

Since  the  interest  of  $  1  at  1  per 

$  6.0  0  )  4  8.0  0  (  8  per  cent.      cent,  for  2  years  is  2  cents,  the  in- 

4  8.0  0  terest  of  S  JOO  will  bo  300  times  as 

much,  or  $  G.     Now,  if  S  6  is  1  per 

cent.,  S  48  will  be  as  many  per  cent,  as  $j  G  is  contained  times  in 
$48,  or  8  per  cent. 

Rule.  —  Divide  the  given  interest  hy  the  interest  of  the  given  sum  at  1 
per  cent,  for  the  given  time,  and  the  quotient  will  be  the  rate  per  cent, 
required. 

202.  What  is  a  problem  ?  —  203.  How  many  tonus  or  thintrj!  have  been 
fiiven  in  the  preceding  questions  in  interest  ?  Name  thoin.  What  does  an 
investigation  of  these  terms  involve  ?  Name  them.  How  many  terms  nro 
frivon  in  each  pmhlcm  in  order  to  find  a  fourth  ?  What  \\\o  ])rol)l('nis  iiave 
liiM'n  examined?  —  204.  Wliat  is  prol)lem  III.?  Exi)Uiin  the  o))cration. 
Th(!  rule  for  finding  the  rate  per  cent.,  the  principal,  interest,  and  thao  being 
given  i 


PROBLEMS  IN  INTEREST.  211 

Examples  for  Practice. 

2.  The  interest  of  $250  for  1  year,  3  months,  is  $28,125; 
what  is  the  rate  per  cent.  ?  Ans.  9  per  cent. 

3.  If  I  pay  $  8.82  for  the  use  of  $  72  for  1  year,  9  months, 
what  is  the  rate  per  cent.  ?  Ans.  7  per  cent. 

4.  A  note  of  $  500,  being  on   interest    2    years,    6   months, 
amounted  to  $  550 ;  what  was  the  rate  per  cent.  ? 

Ans.  4  per  cent. 

5.  The  intei-est  for  $  700  for  1  year,  G  months,  is  $  63 ;  what 
is  the  rate  per  cent.  ?  Ans.  6  per  cent. 

6.  If  I  pay  $  53.78^  for  tlie  use  of  $  922  for  1  year,  2  months, 
what  is  the  rate  per  cent.  ?  Ans.  5  per  cent. 

205.    Problem  TV.     To  find   the  time,  the  principal, 
interest,  and  rate  per  cent,  being  given. 

Ex,  1.  For  how  long  a  time  must  $  300  be  on  interest  at  6 
per  cent,  to  gain  $  36  ?  Ans.  2  years. 

OPERATION.  We   find   the    interest  on   the 

$3  0  0  given    principal    for    1    year,   by 

^0  6  which   we   divide    the   given   in- 

•  terest. 

$  1  8.0  0)  3  6.0  0  (  2  years.  since  the  interest  of  $  1  for  1 

3  6.0  0  year   is   6    cents,  the   interest   of 

"7"^  _  S  300  will  be  300  times  as  much, 

or  S  18.     Now,  if  it  require  1  year  for  the  given  principal  to  gain  $18, 

it  will  require  as  many  years  to  gain  $  36  as  $  18  is  contained  times  in 

$  36,  or  2  years. 

Rule.  —  Divide  the  given  interest  by  tlie  interest  of  the  given  principal 
for  1  year,  and  the  quotient  will  be  the  time. 

Examples  for  Practice. 

2.  If  the  interest  of  $  140  at  6  per  cent,  is  $  42,  for  how  long 
a  time  was  it  on  interest  ?  Ans.  5  years. 

3.  How  long  a  time  must  $  165  be  on  interest  at  6  per  cent, 
to  gain  $  14.85  ?  Ans.  1  year,  6  months. 

4.  How  long  must  $  98  be  on  interest  at  8  per  cent,  to  gain 
$  25.48  ?  Ans.  3  years,  3  months. 

5.  A  note  of  $  680  being  on  interest  at  4  per  cent,  amounted 
to  $  727.60  ;  how  long  was  it  on  interest  ? 

Ans.  1  year,  9  months. 

205.  What  is  Problem  IV.  ?     Exphxin  the  operation.     The  rule  for  find, 
iug  the  time,  the  principal,  interest,  and  rate  per  cent,  being  given  i 


212  COMPOUND  INTEREST. 

206.  Problem  Y.  To  find  the  peincipal,  the  inter- 
est, time,  and  rate  per  cent,  being  given. 

Ex.  1.  What  principal  at  6  per  cent,  will  gain  $  36  in  2 
years  ?  Ans.  $  300. 

OPEBATION.  We  find  the  interest  of  S  1 

.0  G  int.  of  $  1  for  1  y.  ^o^  2  years,  by  which  we  divide 

o  the  given  interest. 

. Since  it  requires  2  years  for 

.12)$3  6.0  0($300  principal.       a  principal  of  S  1   to  gain  12 

cents,  it  will  require  a  principal 
of  as  many  dollars  to  gain  $  36  as  $  0.12  is  contained  times  in  S  36, 
or  $  300. 

Rule.  —  Divide  the  given  interest  or  amount  by  the  interest  or  amount 
of  $1  for  the  given  rate  and  time,  and  the  quotient  will  be  the  principal. 

Examples  for  Practice. 

2.  What  principal  will  gain  $  24.225  in  4  years,  3  months, 
at  6  per  cent.  ?  Ans.  $  95. 

3.  What  principal  will  gain  $  5.11  in  3  years,  6  months,  at  8 
per  cent.  ?  Ans.  $  18.25. 

4.  The  interest  on  a  certain  note  at  9  per  cent,  in  1  year 
and  8  months  amounted  to  $  42 ;  what  was  the  full  amount 
of  the  note  ?  Ans.  $  280. 


COMPOUND    INTEREST. 

.    207.    Compound    Interest    is    interest    on    both    principal    and 
interest,  when  the  latter  is  not  paid  on  becuming  due. 

The  law  specifies  that  the  borrower  of  money  shall  pay  the 
lender  a  certain  sum  for  the  use  of  $  100  for  a  year.  Now,  if  he 
does  not  pay  this  sum  at  the  end  of  the  year,  it  is  no  more  than 
ju.~t  that  he  should  pay  interest  for  the  us;e  of  it  as  lonijj  a>  he 
shall  keep  it  in  his  possession.  The  computation  of  compound 
interest  is  based  upon  this  principle. 

206.  What  is  Problem  V.  ?  Explsiin  the  opcrntion.  The  rule  far  findinc; 
the  pnnn])al,  the  interest,  time,  niul  rate  per  cent,  beinq:  given? —  207. 
What  is  compound  interest  ?     On  what  principle  is  it  based  1 


COiMPOUND   INTEREST.  213 

208.   To  find  the  compound  interest  of  any  sum. 

Ex.  1.   What  is  the  compound  interest  of  $500  for  3  years,  7 
months,  and  12  days,  at  G  per  cent.  ?  Ans.  $  117.541. 

OPERATION. 

Principal,  $  5  0  0 

Interest  of  $  1  for  1  year,  ,0  6 

Interest  for  1st  year,  3  0.0  0 

500 


Amount  for  1st  yeai-,  5  3  0.0  0 

.0  6 

Interest  for  2d  year,  3  1.8  0  0  0 

5  3  0.0  0 

Amount  for  2d  year,  5  6  1.8  0 

.0  6 

Interest  for  3d  year,  3  3.7  0  8  0 

5  6  1.8  0 

Amount  for  3d  year,  5  9  5.5  0  8 

Interest  of  $  1  for  7mo.  12  da.,  .0  3  7 

4.1  6  8  5  5  6 
1  7.8  6  5  2  4 


Interest  for  7mo.  12da.,  2  2.0  3  3  7  9  6 

5  9  5.5  0  8 

Amount  for  3y.  7mo.  12da.,  6  1  7.5  4  1  7  9  6 

Principal  subtracted,  5  0  0 

Compound  interest,  $11  7.5  41796 

We  first  find  tlie  interest  of  tbe  principal  fi)r  1  year,  and  add  the 
interest  to  the  principal  for  a  new  principal.  We  then  find  the  inter- 
est of  this  principal  for  1  year,  and  proceed  as  before ;  and  so  also 
with  the  third  year.  For  the  months  and  days  we  find  the  interest  on 
the  amount  for  the  last  year,  and,  adding  it  as  before,  we  subtract  the 
original  principal  from  the  last  amount  for  the  answer. 

Rule.  —  Find  the  interest  of  the  given  sum  for  one  year,  and  add  it  to 
the  principal;  then  find  the  amount  of  this  amount  for  the  next  year ;  and 
so  continue,  until  the  time  of  settlement. 

If  there  are  months  and  days  in  the  given  time,  find  the  amount  for 
them  on  the  amount  for  the  last  year. 

Subtract  the  principal  from  the  last  amount,  and  the  remainder  is  the 
compound  interest. 

208.  Explain  the  operation  in  computing  compound  interest.    The  rule  ? 


214 


COMPOUND   INTEREST. 


Note  1.  —  If  the  interest  is  to  be  paid  semi-annually,  quarterly,  monthly, 
or  daily,  it  must  be  computed  for  the  half-3'ear,  quarter-year,  month,  or  day, 
and  added  to  the  principal,  and  then  the  interest  computed  on  this,  and  on 
each  succeeding  amount  thus  obtained,  up  to  the  time  of  settlement. 

Note  2.  —  When  partial  payments  have  been  made  on  notes  at  com- 
pound interest,  the  rule  is  like  that  adopted  in  Art.  199. 

Examples  for  Practice. 

2.  What  is  the  compound  interest  of  $ 761.75  for  4  years? 

Ans.  $  199.941. 

3.  What  is  the  amount  of  $  67.25  for  3  years,  at  compound 
interest  ?  Ans.  $  80.095. 

4.  What  is  the  amount  of  $  78.69  for  5  years,  at  7  per  cent.  ? 

Ans.  $  110.364. 

5.  What  is  the  amount  of  $  128  for  3  years,  5  months,  and 
18  days,  at  compound  interest?  Ans.  $  156.717. 

6.  What  is  the  compound  interest  of  $  76.18  for  2  years,  8 
months,  9  days  ?  .  Ans.  $  12.967. 

200.  Method  of  computing  compound  interest,  by 
means  of  a 

TABLE 

Showing  the  Amount  of  $  1,  or  £  1,  for  ant  Number  of  Years,  from 
1  to  20,  at  3,  4,  5,  6,  AND  7  per  cent.,  Compound  Interest. 


Years. 

3  per  cent. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

Years. 

1 

1.030000 

1.040000 

1.050000 

1.060000 

1.070000 

1 

2 

1.060900 

1.081600 

1.102500 

1.123600 

1.144900 

2 

3 

1092727 

1.124864 

1.157625 

1.191016 

1.225043 

3 

4 

1.125.508 

1.169858 

1.21.5506 

1.262476 

1.310796 

4 

5 

1.159274 

1.216652 

1.276281 

1.338225 

1.402552 

5 

6 

1.194052 

1.265319 

1.340095 

1.418519 

1  500730 

6 

7 

1.229873 

1.315931 

1.407100 

1.5036.30 

1.605781 

7 

8 

1.266770 

1.368569 

1.477455 

1.593848 

1.71818G 

8 

9 

1..'304773 

1.42.3311 

1.551328 

1.689478 

1.838459 

9 

10 

1..343916 

1.480244 

1.628894 

1.790847 

1.967151 

10 

11 

1.384233 

1., 539454 

1.7103.39 

1.898298 

2.104852 

11 

12 

1.425760 

1.601032 

1.795856 

2,012196 

2.252191 

12 

1.3 

1.4685.33 

1.665073 

1.885649 

2.132923 

2.409845 

13 

14 

1.512589 

1.731676 

1.979931 

2.260903 

2.578534 

14 

1.5 

1.557967 

1.800943 

2.078928 

2.396558 

2.750032 

15 

16 

1.604706 

1.872981 

2.182874 

2.540351 

2.952164 

16 

17 

1.652847 

1.947900 

2.292018 

2.692772 

3.158815 

17 

18 

1.702433 

2.025816 

2.406619 

2.854339 

3  379932 

18 

19 

1.753.506 

2.106849 

2.526950 

3.025599 

3.616527 

19 

20 

1.806111 

2.191123 

2.6.53297 

3.207135 

3.869685 

20 

209.  If  the  interest  is  to  bo  paid  semi-annually,  quarterly,  &c.,  how  is  it 
computed  ?     How,  when  partial  payments  have  been  made  t 


COMPOUND   INTEREST.  215 

Ex.  1.   What  is  tlie  interest  of  $  240  for  6  years,  4  months, 
and  G  days,  at  6  per  cent.  ?  Ans.  $  107.593. 

OPEKATION. 

Amount  of  $  1  for  6  years,                              1.4  1  8  5  1  9 
Principal,  ,    240 

56740760 
2837038 


Amount  of  principal  for  6  years,                  3  4  0.4  4  4  5  6  0 
Interest  of  $  1  for  4mo.  6da.,  .         .0  2  1 

34044456 
6808891  2 


Interest  of  amount  for  4mo.  6da.,  7. 14933576 

Amount  added,  3  4  0.4  4  4  5  6  0 

Amount  for  6y.  4rao.  6da.,  34  7.5  9389576 

Principal  subtracted,  2  4  0 

Interest  for  given  time,  $10  7.5  9389576 

"We  multiply  the  principal  by  the  amount  of  S  1  for  6  years  in  the 
table,  and  obtain  the  amount  for-  6  years.  We  then  find  the  interest 
on  this  amount  for  the  4  months  and  6  days,  and  add  it  to  its  princi- 
pal, and  from  the  sum  subtract  the  pi-incipal  for  the  answer.     Hence, 

Multiply  the  amount  of  %1  for  the  given  rate  and  time,  as  found  in  the 
table,  by  the  principal,  and  the  product  loill  be  the  amount.  Subtract  the 
principal  from  the  amount,  and  the  remainder  will  be  the  compound  interest. 

If  there  are  months  and  days  in  the  time,  cast  the  interest  for  the  months 
and  days  as  in  the  foregoing  rule. 

Examples  for  Practice. 

2.  What  is  the  interest  of  $  884  for  7  years,  at  4  per  cent.  ? 

Ans.  $  279.283. 

3.  What  is  the  interest  of  $  721  for  9  years,  at  5  per  cent.  ? 

Ans.  $397,507. 

4.  Wliat  is  the  amount  of  $960  for  12  years,  6  months,  at  3 
percent.?  Ans.  $  1389.26. 

5.  What  is  the  amount  of  '$  25.50  for  20  years,  2  months,  and 
12  days,  at  7  per  cent.  ?  Ans.  $  100.058. 

6.  Wliat  is  the  amount  of  $  12  for  6  months,  the  interest  to  be 
added  each  month?  Ans.  $  12.364 -|-- 

7.  What  is  the  amount  of  $  100  for  6  days,  the  interest  to  be 
added  daily  ?  Ans.  $  100.10004. 


216  DISCOUNT. 


DISCOUNT. 

210t    Discount  is  an  allowance  or  deduction  for  the  payment 

of  a  debt  before  it  is  due. 

* 

The  Present  Worth  of  any  sum  is  the  principal,  which,  being  put 
at  interest,  will  amount  to  the  given  sum  in  the  time  for  which 
the  discount  is  made.  Thus,  $  100  is  the  present  worth  of  $  106, 
due  one  year  hence  at  6  per  cent. ;  for  $  100  at  G  per  cent,  will 
amount  to  $  106  in  this  time ;  and  $  6  is  the  discount. 

Note.  —  Business  men,  however,  often  deduct  five  per  cent.,  or  more, 
from  the  face  of  a  bill  due  in  six  months,  or  a  percentage  greater  than  the 
legal  rate  of  interest. 

211.  The  interest  of  any  sum  cannot  properly  be  taken  for 
the  discount ;  for  the  interest  for  one  year  is  tlie  iiactional  part  of 
the  sum  at  interest,  denoted  by  the  rate  per  cent,  for  the  numera- 
tor, and  100  for  the  denominator  ;  and  the  discount  for  one  year 
is  the  fractional  part  of  the  sum  on  which  discount  is  to  be  made, 
denoted  by  the  rate  per  cent,  for  the  numerator,  and  100  plus  the 
rate  per  cent,  for  the  denominator.  Thus,  if  the  rate  per  cent, 
of  interest  is  6,  the  interest  for  one  year  is  y^jj  of  the  sum  at  in- 
terest ;  but  if  the  rate  per  cent,  of  discount  is  6,  the  discount  for 
one  year  is  y§^  of  the  sum  on  which  discount  is  made. 

212.  In  discount,  the  rate  per  cent.,  time,  and  the  sum  on  which 
the  discount  is  made,  are  given  to  find  the  present  worth. 

These  terms  correspond  precisely  to  Problem  VI.  in  interest, 
in  which  the  rate  per  cent.,  time,  and  amount  are  given  to  find  the 
principal.     (Art.  203.) 

213.  To  find  the  present  worth  and  the  discount  of 
any  sum  due  at  a  future  time. 

Ex.  1.  What  is  the  present  worth  of  $  25.44,  due  one  year 
hence,  discounting  at  6  per  cent.  ?     "NVhat  is  the  discount  ? 

Ans.  %  24  present  worth  ;  $  1.44  discount 

210.  "What  is  discount ?  The  present  worth  of  any  sum  of  money? 
ITow  illustrated  ?  — 211.  Are  interest  and  discount  the  same  ?  Explain  the 
difference.  Which  is  the  greater,  the  interest  or  discount  on  any  sum,  for  a 
given  time?  —  212.  What  terms  are  given  in  discount,  and  what  is  required  ? 
To  what  do  these  correspond  in  interest  T 


DISCOUNT.  217 

OrEHATIOX. 

Amount  of  $  1,         1.0  6  )  2  5.4  4  (  $  2  4,  present  worth. 

212 


4  2  4     $  2  5.4  4,  given  sum. 
4  2  4        2  4.0  0,  present  worth. 

$  1.4  4,  discount. 

Since  the  present  worth  of  S  1.06,  due  one  year  hence",  at  6  per  cent, 
is  S  1,  the  present  worth  of  S  25.44  is  as  many  dollars  as  S  1.06  is  con- 
tained times  in  $  25.44,  or  $  21.  Wc  thus  find  the  present  worth  to  be 
S  24,  which,  subtracted  from  the  given  sum,  gives  $  1.44  as  the  discount. 

Rule.  —  Find  the  amount  o/"  S  1  for  the  given  time  and  rate ;  hy 
which  divide  the  given  sum,  and  the  quotient  will  he  the  present  wouth. 

The  present  toorth  subtracted  from  the  given  sum  will  give  the  dis- 
count. 

Note.  —  The  discount  may  bo  found  directly  by  making  the  interest  of 
$  1  for  the  given  rate  and  time  the  numernlor  of  a  fraction,  and  the  amount 
of  $  1  for  the  given  rale  and  time  the  denominator,  and  then  multiply  the 
given  sum  by  this  fraction. 

Examples  for  Practice. 

2.  "What  is  the  present  worth  of  $  152.64,  due  1  year  hence? 

Ans.  $  144. 

3.  "What  is  the  present  worth  of  $477.71,  due  4  years  hence? 

Ans.  %  385.25. 

4.  What  is  the  discount  of  $  172.86,  due  3  years,  4  months 
hence?  Ans.  $28.81. 

5.  What  is  the  discount  of  $  800,  due  3  years,  7  months,  and 
18  days  hence?  Ans.  $  143.186. 

6.  Samuel  Heath  has  given  his  note  for  S  375.75,  dated  Oct. 
4,  1852,  payable  to  John  Smith,  or  order,  Jan.  1,  1854;  what  is 
the  real  value  of  the  note  at  the  time  given  ?     Ans.  $  349.697. 

7.  Bought  a  chaise  and  harness  of  Isaac  Morse  for  $  125.75, 
for  Avhich  I  gave  him  my  note,  dated  Oct.  5,  1852,  to  be  paid 
in  6  months;  what  is  the  present  value  of  the  note,  Jan.  1, 
1853?  Ans.  $123.81. 


213.  Explain  the  operation  for  finding  the  present  worth  and  discount. 
The  reason  of  die  operation  ?     The  rule  f   What  other  method  is  given  ? 
19 


218  COMillSSION   AiSID   BROKERAGE. 


COMMISSION,    BROKERAGE,    AND    STOCKS. 

214.  Commission  is  the  percentage  paid  to  an  agent,  factor,  or 
commission  merchant,  for  buying  or  selling  goods,  or  transacting 
other  business. 

Brokerage  is  the  percentage  paid  to  a  dealer  in  money  and 
stocks,  called  ^  broker,  for  making  exchanges  of  money,  nego- 
tiating different  kinds  of  bills  of  credit,  or  transacting  other  hke 
business. 

Stocks  is  a  general  name  given  to  government  bonds,  and  to 
the  money  capital  of  corporations,  such  as  banks,  insurance,  rail- 
road, manufacturing,  and  mining  companies. 

Stocks  are  usually  divided  into  equal  shares,  the  market  value 
of  which  is  often  variable. 

When  stocks  sell  for  their  original  value  they  are  said  to  be  at 
par  ;  when  for  more  than  their  original  value,  above  par,  or  at  a 
premium  ;  when  for  less  than  their  original  value,  heloxo par,  or  at 
a  discount. 

The  premium,  or  advance,  and  the  discount  on  stocks,  are 
generally  computed  at  a  certain  per  cent,  on  the  original  value 
of  the  shares. 

The  rate  per  cent,  of  commission  or  brokerage  is  not  regulated 
by  law,  but  varies  in  different  places,  and  with  the  nature  of  the 
business  transacted. 

Commission  and  brokerage  are  computed  in  the  same  manner. 

215.  To  find  the  commission  or  brokerage  on  any  sum. 

Ex.  1.  A  commission  merchant  sells  goods  to  the  amount  of 
$  870  ;  what  is  his  commission  at  3  per  cent.?      Ans.  %  26.37. 

Since  commission  is  a  percentage  on  tlie  given  sum,  tlie  com- 
mission on  $879,  at  3  per  cent.,  will  be  $87'J  X  -03  =  $  20.37. 

RuLK.  —  Find  fJie  percentage  on  the  given  sum  at  the  given  rate  pet 
cent.,  and  the  result  is  the  commission  or  hrokcrage.     (Art.  191.) 

214.  Wlmt  is  coTTiTnis-inn  ?  T5rokorn<re  ?  Stork?  Into  what  nrc  storks 
flividivl  ?  When  lire  storks  iit  par?  Wlicn  al)ovp  pnr  ?  AVIion  hclow  ]inr  ? 
How  is  the  premium  or  disnoimt  on  stocks  comtnitod  ?  IIow  is  cominissioa 
and  brokcra^o  computed  ?  — 215.  What  is  the  rule? 


COMMISSION  AND   BROKERAGE.  219 

Examples  for  Practice. 

2.  Wliat  is  the  commission  on  tlie  sale  of  a  quantity  of  cotton 
goods  valued  at  $  5678,  at  3  per  cent.  ?  Ans.  $  170.34. 

3.  A  commission  merchant  sells  goods  to  the  amount  of  $7896, 
at  2  per  cent. ;  what  is  his  commission  ?  Ans.  $  157.92. 

4.  My  agent  in  Chicago  has  purchased  -wheat  for  me  to  the 
amount  of  $  1728;  what  is  his  commission,  at  1^  per  cent.? 

Ans.  $  25.92. 

5.  My  factor  advises  me  that  he  has  purchased,  on  my  ac- 
count, 97  bales  of  cloth,  at  %  15.50  per  bale  ;  what  is  his  commis- 
sion, at  21  per  cent.  ?  Ans.  $  37.587. 

6.  My  agent  at  New  Orleans  informs  me  that  he  has  disposed 
of  500  barrels  of  flour  at  $  6.50  per  barrel,  88  barrels  of  apples 
at  $2.75  per  barrel,  and  56cwt.  of  cheese  at  $10.60  per  cwt. ; 
what  is  his  commission,  at  3|-  per  cent.  ?  Ans.  $  153.21. 

7.  A  broker  negotiates  a  bill  of  exchange  of  $  2500  at  ^  per 
cent,  commission  ;  what  is  his  commission  ?  Ans.  $  12.50. 

8.  A  broker  in  New  York  exchanged  $  46256  on  the  Canal 
Bank,  Portland,  at  ^  of  1  per  cent. ;  what  did  he  receive  for  his 
trouble  ?  *        Ans.  $  57.82. 

9.  A  broker  in  Baltimore  exchanged  $  20500  on  the  State 
Bank  of  Indiana,  at  ^  of  1  per  cent. ;  what  was  the  amount  of 
his  brokerage  ?  Ans.  $  102.50. 

216.  When  the  given  sum  inchides  botli  the  brokerage 
or  commission  and  tlie  sum  to  be  invested. 

Ex.  1.  A  merchant  in  Cincinnati  sends  $  1500  to  a  commis- 
sion merchant  in  Boston,  with  instructions  to  lay  it  out  in  goods, 
after  deducting  his  commission  of  2^  per  cent. ;  what  is  his  com- 
mission.?  Ans.  $36,586. 

OPERATION. 

$  1500  -^  1.025  =  $  1463.414. 
$  1500 —$  1463.414  =  S  36.586. 

Since  the  agent  is  entitled  to  2^  per  cent,  of  the  amount  he  lays  out,^ 
it  is  evident  he  requires  %  1.021  to  purchase  goods  to  the  amount  of 
S  1.  Hence,  he  can  expend  for  goods  as  many  dollars  as  S  1-02^  is 
contained  times  in  $  15(X).  or  S  14C3.414  ;  Avhich,"being  subtracted  from 
$  1500,  the  amount  sent  him,  leaves  as  his  commission  S  36.586. 

216  How  do  yon  find  tho  rommission  or  brokorafre  when  the  given  sum 
includes  both  tlie  brokerage  or  commission  and  tlie  sum  to  be  invested  ? 


220  COxMMISSION   AND   BROKERAGE. 

Rule.  —  Divide  the  given  sum  hy  1  increased  hy  die  per  cent,  of  com,' 
mission,  and  the  quotient  will  be  the  sum  to  be  invested. 

Subtract  the  sum  to  be  invested  from  the  given  sum,  and  the  remaindef 
will  be  the  commission. 

Examples  for  Practice. 

2.  A  town  agent  has  %  2000  to  invest  in  bank  stock,  after 
deducting  his  commission  of  1  J-  per  cent. ;  what  will  be  his  com- 
mission, and  what  the  sum  invested  ? 

Ans.  $  29.557  commission  ;  $  1970.443  sum  invested. 

3.  A  shoe-dealer  sends  $5256  to  his  agent  in  Boston,  which 
he  wishes  him  to  lay  out  for  shoes,  reserving  his  commission  of  3 
per  cent. ;  what  is  his  commission  ?  Ans.  %  153.088. 

4.  A  broker  expends  $3865.94  for  merchandise,  after  deduct- 
ing his  commission  of  4  per  cent. ;  what  was  his  commission,  and 
what  sum  did  he  expend  ? 

Ans.  $148.69  commission;  $3717.25  sum  expended. 

5.  I  have  sent  to  my  agent  at  Buffalo,  N.  Y.,  $  10000,  to  be 
expended  in  flour,  after  deducting  his  commission  of  3^-  per  cent. ; 
what  will  be  his  commission,  and  the  value  of  the  flour  puz-chased  ? 

Ans.  $  314.76-f-  com. ;  $  9685.23-1-  ^"'^^'  ^^  ^o^r. 

217t  To  find  the  value  of  stocks,  when  at  an  advance 
or  at  a  discount. 

Ex.  1.  What  is  the  value  of  $  2150  railroad  stock,  at  7  per 
cent,  advance  ?  Ans.  $  2300.50. 

OPKIIATION. 

$  2150  X  .07  =  $  150.50  ;  $  2150  +  $  150.50  =  $  2300.50. 

Rule.  —  Find  the  percentage  on  the  given  sum,  and  add  or  subtract, 
according  as  the  stock  is  at  an  advance  or  at  a  discount.     (Art.  191.) 

Examples  for  Practice. 

2.  What  must  be  given  for  10  shares  in  the  Boston  and  IM^ine 
Railroad,  at  15  per  cent,  advance,  the  shares  being  $  100  each  ? 

Ans.  $1150. 

3.  What  must  be  given  for  75  shares  in  the  Lowell  Railroad, 
at  25  per  cent,  advance,  the  original  shares  being  $  100  each? 

Ans.  $  0375. 

216.  What  is  the  rule  ?  —  217.  How  do  you  find  the  value  of  stocks,  when 
at  an  advance  or  at  a  discount  ?     What  is  the  rule  f 


<^ 


BANKING.  221 

4.  "What  is  the  purchase  of  $8979  bank  stock,  at  12  per  cent, 
advance?  Ans.  $  lUUoG.-AS. 

5.  AVhat  is  the  purcliase  of  $  1789  bank  stock,  at  9  per  cent, 
below  par  .^  Ans.  $  1627.99. 

G.  A  stockholder  in  the  Illinois  Central  Railroad  sells  his  right 
of  pui'cha^e  on  5  shares  of  $  100  each  at  12  per  cent,  advance; 
what  is  the 'premium  ?  Ans.  $  60. 

7.  What  is  the  value  of  20  shares  canal  stock,  at  12^  per  cent, 
discount,  the  original  shares  being  $  100  each.        Ans.  $  1750. 

8.  What  is  the  value  of  15  shares  in  the  Livingston  County 
Bank,  at  8|-  per  cent,  advance,  the  original  shares  being  $  100 
each?  Ans.  $  1623.75. 

9.  Bought  87  shares  in  a  certain  corporation,  at  12  per  cent, 
below  par,  and  sold  the  same  at  19^  per  cent,  above  par;  what 
sum  did  I  gain,  the  original  shai'es  being  $  176  each  ? 

Ans.  $  4795.87i. 


BANKING 


218.  A  Bank  is  a  joint  stock  company,  established  for  the 
purpose  of  receiving  deposits,  loaning  money,  dealing  in  ex- 
change, or  issuing  bank-notes  or  bills,  as  a  circulating  medium, 
redeemable  in  specie  at  its  place  of  business. 

The  Capital  of  a  bank  is  the  money  paid  in  by  its  stockholders, 
as  the  basis  of  business. 

Banking  is  the  general  business  commonly  transacted  at  banks. 

Note.  —  The  persons  chosen  by  the  stockholders  to  mana,2:e  the  affairs 
of  the  bank  are  called  its  board  of  directors,  who  select  one  of  their  own 
number  as  president,  and  some  person  as  cashier. 

The  president  and  cashier  sign  the  bills  issued,  which  also  are,  in  some 
ij^tances,  countersigned  by  some  State  otHcer. 

The  cashier  superintends  the  bank  accounts ;  and  another  person,  called 
the  teller,  usually  receives  and  pays  out  money. 

A  check  is  an  order  drawn  on  the  cashier  of  the  bank  for  money. 

218.  AVhat  is  a  bank?     The  capital  of  a  bank ?     Banking ?     Who  chooso 
the  directors?     Who  choose  the  president  and  cashier?     Who  sign  the  billa 
issued  ?     Who  superintends  tlie  accounts  1     Who  receives  and  pa^'s  out  tha 
money  ?     What  is  a  check  ? 
19* 


222  BANKING. 

BANK  DISCOUNT. 

219t  Bank  Discount  is  the  simple  interest  of  a  note,  di-aft, 
or  bill  of  exchange,  deducted  from  it  in  advance,  or  before  it 
becomes  due. 

The  interest  is  computed,  not  only  for  the  specified  time,  but 
also  for  three  days  additional,  called  days  of  grace.  Thus,  if  a 
note  is  given  at  the  bank  for  60  days,  the  interest,  which  i^ 
called  the  discount,  is  computed  for  63  days  ;  and  if  the  note 
is  paid  within  this  time,  the  debtor  complies  with  the  require- 
ments of  the  law. 

The  Legal  Rate  of  Discount  is  usually  the  same  as  the  legal  rate 
of  interest ;  and  the  difference  between  hanh  discount  and  true 
discount  is  the  same  as  the  difference  between  intei-est  and  true 
discount. 

A  note  is  said  to  be  discounted  at  a  bank,  when  it  is  received 
as  security  for  the  money  that  is  paid  for  it,  after  deducting 
the  interest  for  the  time  until  it  shall  become  due. 

The  Avails,  Pi'ijcceds,  or  Present  'Wortli  of  a  note  is  the  sum 
paid  for  it. 

220i  To  find  the  bank  discount  and  the  present  worth  of  a 
note. 

Ex.  1.  What  is  the  bank  discount  on  $  842  for  90  days,  at  6 
per  cent.  ?     What  is  the  present  worth  ? 

Ans.  $  13.051  discount ;  $  828.949  present  worth. 

FIUST   OPERATION.  SECOND   OPERATION. 

Sum  discounted,        $842  Sum  discounted,  $842,000 

Int.  of  S 1  for  93d.     .0155  i      r  ■  ^  e     cr\A  oToa 
-jIq- of  sum  =  int.  for  60a.,              8.420 

4210  \    of  int.  for  GOd.  ==  int.  for  30d.,  4.210 

4210  J^  ofint.  fur30d.  =  iut.  forSd.,      .421 

_-  Bank  discount,  $13,051 

Bank  discount,  $13-0510  p^^^^^^  ^^^^^j^^  $i^49 

Present  worth  =  $  842  — $  13.051  =  $  828.949,  Ans. 

We  find  tlio  interest  of  the  sum  discounted,  as  in  Art.  193,  and  tlii.-- 
interest  is  the  bank  discount,  which,  subtracted  from  the  sum  discounted, 
gives  the  proceeds,  or  the  present  worth. 

219.  What  is  bank  discount?  "\V1umi  is  it  paid?  Is  interest  cominitod 
for  more  iliaii  the  speeilird  time?  AViiat  arc  tliesc  three  additional  days 
called  ?  How  will  you  ilhistnite  this  ?  What  is  the  letral  rate  of  diseouiu  ? 
Tlio  diffcrenee  between  bank  discount  and  true  discount?  When  is  a  note 
said  to  1)C  discounted  at  a  bank  ?  What  is  the  sum  ])aid  for  it  called  ?  —  220. 
Explain  the  operation  for  finding  the  bank  discount  on  any  sum. 


BANK  DISCOUNT.  223 

RuT-E.  —  Fltid  the  interest  on  the  note,  or  sum  discounted,  for  the 
given  rale  and  tune,  including  three  days  of  grace,  and  this  interest  is 
the  DISCOUNT. 

Subtract  the  discount  from  the  face  of  the  note  or  sum  discounted,  and 
the  remainder  is  the  presknt  worth. 

Note.  —  A  convenient  method  of  calculating  interest  for  days,  is  to 
divide  the  principal  by  100,  by  removing  the  decimal  point  two  places  to  the  left, 
and  then  taking  such  a  part  of  the  quotient  as  the  given  number  of  days  is  part 
of  60  days  ;  as  iu  the  second  operation. 

Examples  for  Practice. 

2.  What  is  the  bank  discount  on  $  478  for  60  days  ? 

Ans.  $5,019. 

3.  What  is  the  bank  discount  on  $  780  for  30  days  ? 

Ans.  $  4.29. 

4.  Wliat  is  the  bank  discount  on  $  1728  for  90  days  ? 

Ans.  $  26.784. 

5.  How  much  money  should  be  received  on  a  note  of  $  1000, 
payable  in  4  mouths,  discounting  at  a  bank  where  the  interest  is 
6  per  cent.  ?  Ans.  $  979.50. 

6.  What  sum  must  a  bank  pay  for  a  note  of  $  875.35,  payable 
in  7  months  and  15  days,  discounting  at  7  per  cent.  ? 

Ans.  S  836.542. 

7.  What  are  the  avails  of  a  note  of  $  596.24,  payable  in  8 
months  and  9  days,  discounted  at  a  bank  at  8  per  cent.  ? 

Ans.  $  562.85. 

8.  What  is  the  bank  discount  of  a  draft  of  $  1350.50,  payable 
in  1  year,  4  months,  at  5  per  cent.?  Ans.  $90,596. 

221.  To  find  the  amount  for  which  a  note  must  be 
given,  that  the  avails  may  be  a  specified  sum. 

Ex  1.  For  what  amount  mast  a  note  be  given,  payable  in  90 
days,  to  obtain  $  500  from  a  bank,  discounting  at  6  per  cent.  ? 

Ans.  $  507.872. 

opEEATioN.  Since  %  0.984.5,  present  worth, 

$  1.0  0  0  0      requires  Si  to  be  discounted  for 

Int.  of  $  1  for  93da.,        .0155      the  given  time,  $  500  will  require 

-p„„^„,.^^ a,^e<3;t  (\  Q  A  K  3S  many  dollars  to  be  discounted 

Irresent  worth  or  $1,       .9845  „„  jtAno^r   •         :  •      i  *•    „„  ;„ 

«E -nn         oQ<-        i^  ni\-!  oT,  ^^  ©0.9845   is  contained  tunes  in 

^OUU -f-.J«40  =  ^507.872  $  500,  or  S  507.8  72.     Hence  the 

220.  The  rule?  —  221.  Explain  tlie  operation  for  findincf  the  amount  for 
which  a  note  must  be  given  at  a  bank  to  obtain  a  specified  sum  for  a  given 
time. 


224  INSURANCE. 

Rule. —  Divide  the  given  sum  hy  the  present  tvorth  of  $1  for  the 
given  rate  of  bank  discount  and  time,  including  three  days  of  grace, 
and  the  quotient  will  be  the  answer. 

Examples  for  Practice. 

2.  For  what  sum  must  I  give  my  note  at  a  bank,  payable  in 
4  months,  at  6  per  cent,  discount,  to  obtain  $  300  ? 

Ans.  $306,278. 

3.  A  merchant  sold  a  quantity  of  lumber,  and  received  a  note 
payable  in  6  months ;  he  had  his  note  discounted  at  a  bank,  at  6 
per  cent.,  and  received  $  4572.40.  What  was  the  amount  of  his 
note?  Ans.  $4716.245. 

4.  A  gentleman  wishes  to  take  $  1000  from  the  bank  ;  for 
what  sum  must  he  give  his  note,  payable  in  5  months,  at  6  per 
cent,  discount  ?  Ans.  $  1026.167. 

5.  The  avails  of  a  note,  discounted  at  the  bank  for  8  months, 
at  7^  per  cent.,  were  S  483.56  ;  what  was  the  face  of  the  note  ? 

Ans.  $  509.345. 


INSURANCE. 

222.  Insurance  is  indemnity  obtained,  by  paying  a  certain  sum, 
against  such  losses  of  property  or  of  hfe  as  are  agreed  upon. 

The  Insurer  or  Underwriter  is  the  party  taking  the  risk,  and  the 
Insured  the  party  protected. 

The  Policy  is  the  written  obligation,  or  contract,  entered  into 
between  the  parties. 

Premium  is  the  amount  of  percentage  paid  on  the  property  in- 
sured ibr  one  year,  or  any  specified  time. 

As  a  security  against  fraud,  property  is  not  u:=ual]y  insured 
for  its  whole  value,  nor  is  the  insurer  or  underwriter  bound  to 
indemnify  the  insured  for  a  loss  more  than  is  specified  in  the 
j)o]icy. 

221.  Wlmt  i^  tlie  rule  ?  — 222.  Wliat  is  insurance?  "What  is  the  vn^-tv 
vn\\f<\  that  takes  the  risk  f  What  is  the  party  called  tluit  is  protected  ? 
Wiiat  is  the  policy  1  The  premium  ?  Is  property  usually  insured  to  its 
whole  value  ■? 


CUSTOM-HOUSE   BUSINESS.  225 

223.   To  find  the  premium,  the  rate  and  amount  being 
given. 

Ex.  1.   What  is  the  premium  on  $  485  at  2  per  cent.  ? 

Ans.  $  9.70. 

OPERATION. 

$  485  X  -02  =  $  9.70, 

Rule. —  Find  the  percentage  on  the  given  sum,  and  the  result  is  Oie 
premium.     (Ai-t.  191.) 

Examples  for  Practice. 

2i  What  is  the  premium  on  $  8G8  at  12  per  cent.  ? 

Ans.  $  104.16. 

3.  What  is  the  premium  on  $  1728  at  15  per  cent.  ? 

Ans.  $  259.20. 

4.  A  house,  valued  at  $  3500,  is  insured  at  If  per  cent ;  what 
is  the  premium  ?  Ans.  $  G1.25. 

5.  A  vessel  and  cargo,  valued  at  $  35000,  are  insured  at  3f  per 
cent. ;  now,  if  this  vessel  should  be  destroyed,  what  will  be  the 
actual  loss  to  the  insurance  company?  Ans.  $33687.50. 

6.  A  cotton  factory  and  its  machinery,  valued  at  $  75000,  are 
insured  at  2^  per  cent. ;  what  is  the  yearly  premium  ?  and  if  it 
should  be  destroyed,  what  loss  would  the  insurance  company 
Bustain.''  Ans.  $1875  premium;  $73125  loss. 


CUSTOM-HOUSE    BUSINESS. 

224.  Duties  are  sums  of  money  required  by  government  to  be 
paid  on  imported  goods. 

All  goods  from  foreign  countries  brought  into  the  United  Slates 
are  required  to  be  landed  at  particular  places,  called  ports  of 
entry,  where  are  custom-houses,  at  which  the  duties  or  revenue 
is  collected. 

Duties  are  either  specific  or  ad  valorem. 

A  Specific  Duty  is  a  certain  sum  paid  onr a  ton,  hundred  weight, 
yard,  gallon,  &c 

223.  What  is  the  rule  for  finding  the  premium  on  any  nmoiint  of  property 
insured  ?  —  224.  What  are  duties  ?  Where  are  duties  collected  i  What  is  a 
specific  duty  ? 


226  CUSTOJI-JIOUSE   BUSINESS. 

An  Ad  valorem  Duly  is  a  certain  per  cent,  paid  on  the  actual  cost 
of  the  goods  in  the  country  from  v/hich  they  are  imported. 

Draft  is  an  allowance  for  loaste  made  in  the  weight  of  goods. 

Tare  is  an  allowance  made  for  the  weight  of  the  cask,  box,  &c., 
containing  the  commodity. 

Leakage  is  an  allowance  for  waste  made  on  liquors. 

Gross  Weight  is  the  weight  of  the  commodity,  together  with  the 
cask,  box,  bag,  &c.,  containing  it. 

Net  Weight  is  what  remains  after  all  allowances  have  been  made. 

By  the  tariff  of  ISGlj  duties  are  specific  on  some  articles  ;  and 
on  others,  either  ad  valorem  or  specific  and  ad  valorem. 

It  has  been  decided  that  no  allowances  for  tare,  draft,  break- 
age, &e.,  are  applicable  to  imports  subject  to  ad  valorem  duties, 
except  actual  tare,  or  weight  of  a  cask,  or  package,  and  the 
actual  drainage,  leakage,  or  damage.  The  collector  may  cause 
these  to  be  ascertained,  when  he  has  any  doubts  as  to  what  they 
are. 

225.    To  calculate  ad  valorem  duties. 

Ex.  1.  At  25  per  cent.,  what  is  the  ad  valorem  duty  on  165 
yards  of  broadcloth,  at  $  5  per  yard  ?  Ans.  $  206.25. 

OPERATION. 

$5X165  =  $825;  $825X.2  5=$20  6.2  5,  duty. 

Rule.  —  Find  the  percentage  on  the  cost  of  the  goods,  and  the  result 
is  the  ad  valorem  duty.     (Art.  191.) 

Note.  — When  there  is  actual  draft  or  tare,  the  necessary  deductions  must 
be  made,  before  reckoning  the  duty. 

Examples  for  Practice. 

2.  "What  is  the  duty  on  17281b.  of  copper  sheathing,  invoiced 
at  $  3200,  at  20  per  cent,  ad  valorem  ?  Ans.  $  640. 

3.  What  is  the  duty  on  223111).  of  Russian  iron,  at  30  per 
cent,  ad  valorem ;  the  cost  of  the  iron  being  4  cents  per  lb.  ? 

Ans.  $  26.772,  duty. 

4.  "What  is  the  duty  on  16911b.  of  lead,  at  20  per  cent,  ad 
valorem  ;  the  value  of  the  lead  being  5  cents  per  pound  ? 

Ans.  $16.01,  duty. 

224.  "What  is  an  nd  valorem  duty?  "What  is  drift  ?  Tiire?  Gro<;s  weight  ? 
Net  wciglit? — 225.   What  is  the  rule  for  lindlng  tlic  ad  valorem  duty  1 


ASSESSMENT    OF   TAXES.  227 

5.  Wh.at  is  the  duty  on  10  hogsheads  of  molasses,  each  hogs- 
head gaiighig  150  gallons  gi'oss,  the  actual  wants  being  5  gallons 
to  each  hogshead,  and  the  cost  of  the  molasses  25  cents  per 
gallon  ;  duty  20  per  cent,  ad  valorem  ?         Ans.  $  72.50,  duty. 

G.  What  are  the  net  weight  and  duty,  at  30  per  cent,  ad 
valorem,  on  13  boxes  of  sugar,  weighing  gross  450  pounds  each; 
actual  tare  15  per  cent.,  and  the  cost  of  the  sugar  being  8  cents 
per  ijound  ?  Ans.  4972^  lbs.,  net  weight;  §  119.34,  duty. 

7.  What  is  the  duty  on  an  invoice  of  woollen  goods,  which 
cost  in  Liverpool  1376  £  sterhng,  at  30  per  cent,  ad  valorem; 
the  pound  sterling  being  $  4.84?  Ans.  $  1997.95-j-. 

8.  What  is  the  duty  on  an  invoice  of  goods,  which  cost  in 
Paris  $  2340,  at  80  per  cent,  ad  valorem? 

Ans.  $  1872. 


ASSESSMENT    OF    TAXES. 

22Q,  A  Tax  is  a  sum  of  money  assessed  by  government  for 
public  purposes,  on  property,  and  in  most  States  on  persons. 

Taxes  may  be  either  direct  or  indirect. 

A  Direct  Tax  is  one  imposed  on  the  income  or  property  of  an 
individual. 

An  Indirect  Tax  is  one  imposed  on  the  articles  for  which  the 
income  or  property  is  expended. 

A  Poll  or  Capitation  Tax  is  one  without  regard  to  property,  on 
the  person  of  each  male  citizen,  liable  by  law  to  assessment.  A 
person  so  liable  is  termed  a  poll. 

Real  Estate  is  immovable  propei'ty,  such  as  lands,  houses,  &c. 

Pci'SOnnl  Property  is  all  other  property,  such  as  money,  notes, 
cattle,  furniture,  &c. 

The  method  of  assessing  taxes  is  not  precisely  the  same  in  all 
the  States,  yet  the  principle  is  virtually  the  same. 

The  following  is  the  law  regulating  taxation  in  Massachusetts. 

226.  What  is  a  tax  ?  A  direct  tax  ?  An  indirect  tax  ?  "Wiiat  is  real 
estate?  Personal  property"?  What  is  a  poll  or  capitation  tax?  What  is 
a  poll  ?     Is  the  metliod  of  assessing  taxes  the  same  in  all  the  States  ? 


228  ASSESSMENT   OF   TAXES. 

"  TTie  assessors  shall  assess  ujjon  the  polls,  as  nearly  as  may 
be,  one  sixth  part  of  the  whole  sum  to  he  raised  ;  hut  the  whole  poll 
tax  assessed  in  any  one  year  upon  any  individual  for  toxvn,  county, 
and  state  puiposes,  except  highway  taxes  separately  assessed,  shall 
not  exceed  two  dollars  ;  and  the  residue  of  such  whole  sum  to  he 
raised  shall  he  apportioned  upon  property  ;  "  that  is,  on  the  real 
and  personal  estate  of  individirals  ivhich  is  taxahle.  (General 
Statutes,  p.  78,  as  amended,  1862.) 

227.   To  assess  a  town  or  other  tax.         * 

Ex.  1.  The  tax  to  be  assessed  on  a  certain  town  is  $  2200. 
The  real  estate  of  the  town  is  valued  at  $  60000,  and  the  per- 
sonal property  at  S  30000.  There  are  400  polls,  each  of  wliioh 
is  taxed  $  1.00.  What  is  the  tax  on  $  1.00?  What  is  A's  tax, 
whose  real  estate  is  valued  at  $  2000,  and  his  personal  property 
at  §  1200,  and  who  pays  for  2  polls .'' 

OPEUATION. 

$  1.00  X  400  =  $  400,  amount  assessed  on  the  polls. 

$  2200  —  $  400  =  $  1800,  amount  to  be  assessed  on  the  property. 

$  60000  4-  $  30000  =  $  90000,  amount  of  taxable  property. 

$  1800  -J-  $  90000  =  $  0.02,  tax  on  $  1.00. 

$  2000  X  .02  =  $  40,  A's  tax  on  real  estate. 

$  1200  X  -02  =  $  24,  A's  tax  on  personal  property. 

$  1.00  X  2  =  $  2,  A's  tax  on  2  polls. 

$  40  +  $  24  +  $  2  =  $  66,  amount  of  A's  tax. 

Hence,  in  assessing  taxes,  it  is  necessary  to  have  an  inventory  of  the 
taxable  property,  and,  if  a  levy  on  the  polls  is  to  be  included,  there 
should  be  also  a  complete  bst  of  taxable  polls.     Having  these,  we  then 

Multiply  the  tax  on  each  poll  by  the  number  of  taxable  polls,  and  the 
product  subtracted  from  the  whole  sum  to  be  raised,  will  give  the  sum 

TO    BK    RAISED    OX    THE    PKOPEUTY. 

Tiie  sum  to  be  raised  on  properly  divided  by  (he  tchole  taxable  property, 
will  give  the  sum  to  be  paid  on  each  dollar  of  property  taxed. 

Jilach  man's  taxable  j)roperty,  multiplied  by  the  number  denoting  the 
sum  to  be  paid  on  Si,  with  his  poll  tax  added  to  the  product,  will  give 

THE    amount   of   his    TAX. 

Examples  for  Practice. 

2.  The  town  of  L  is  taxed  S  3600.     The  real  estate  of  the 

town    is    valued    at    $  560,000,    and    the    jiorsonal    property   at 

$  152,500.     There  arc  600  polls,  each  of  which  is  taxed  $  1.25. 

AVliat  is  tlie  per  cent,   or  tax  on  $1.00?  and  what   is  B's  tax. 


22fi.  The  Inw  rcyukting  taxation  in  Massachusetts?  —  227.  TIio  riilu  lor 
assessing  taxes  ? 


ASSESSiMENT   OF   TAXES. 


229 


whose  real  estate  is  valued  at  $4100,  and  his  personal  property 
at  $  1800,  he  paying  lor  lour  polls  ? 

Ans.  $  .004,  tax  on  $  1  ;  $  28.60,  B's  tax. 

3.  "What  is  the  tax  of  a  non-resident,  having  property  in  the 
same  town,  worth  $  15800  ?  Ans.  $  G5.20. 

4.  AVhat  is  D's  tax,  who  pays  for  3  polls,  and  whose  real 
estate  is  valued  at  $40000,  and  his  personal  property  at  $  23G00.'' 

Ans.  $  258.15. 

228.  The  assessing  of  taxes  may  be  facilitated  by  the  use  of  a 
table,  which  can  be  easily  made  after  finding  the  tax  on  $  1. 

Ex.  1.  A  tax  of  $  3900  is  to  be  assessed  on  the  town  of  P. 
The  real  estate  is  valued  at  $  840000,  and  the  personal  property 
at  $  210000  ;  and  there  are  500  polls,  each  of  which  is  taxed 
$  1.50.     What  is  the  assessment  on  $  1  ?  Ans.  $  .003. 

Having  found  the  tax  on  $  1  to  be  $  .003,  before  proceeding 
to  make  the  assessment  on  the  inhabitants  of  the  town,  we  find 
the  tax  on  $  2,  $  3,  &c.,  and  arrange  the  numbers  as  in  the  fol- 
lowing 

TABLE. 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$0,003 

$20 

$0.06 

$300 

$0.90 

$  4,000 

$12.00 

2 

0.006 

30 

0.09 

400 

1.29 

5,000 

15.00 

3 

0.009 

40 

0.12 

500 

1.50 

6,000 

18.00 

4 

0.012 

.50 

0.15 

600 

1.80 

7,000 

21.00 

5 

0.01.5 

60 

0.18 

700 

2.10 

8,000 

24.00 

6 

0.018 

70 

0.21 

800 

2.40 

9,000 

27.00 

7 

0.021 

80 

0.24 

900 

2.70 

10,000 

30. 00 

8 

0.024 

90 

0.27 

1 ,000 

3.00 

20,000 

60.U0 

9 

0.027 

100 

0.30 

2,000 

600 

30,000 

90.00 

10 

0.030 

200 

0.60 

3,000 

9.00 

40,000 

120.00 

2.  What  is  E's  tax,  by  the  table,  whose  property  is  valued  at 


$  1860,  and  who  pays  3  polls  ? 


OPERATION. 

Tax  on 

$1000 

is 

$3.0  0 

Si 

<( 

800 

li 

2.4  0 

ii 

(( 

60 

i( 

.18 

a 

a 

3  polls 

a 

4.5  0 

Valuation,  $18  6  0;    $  1  0.0  8,  Tax. 


Ans.  $  10.08. 

We  find  in  the  table  the  tax 
on  $  1000,  and  then  on  S  800, 
and  then  on  $60,  and  to  these 
sums  add  the  tax  on  the  3 
polls,  at  $1.50  each,  for  the 
answer. 


228.  How  m.iy  the  operation  of  assessing  taxes  be  facilitated  1    How  is 
the  above  table  formed  ? 
20 


230  EQUATION  OF  PAYMENTS. 

3.  Wliat  is  F's  tax,  whose  real  estate  is  valued  at  $  6535,  and 
his  personal  px'opertj  at  $  3175,  and  who  pays  for  6  polls  ? 

Ans.  $  38.13. 

4.  What  is  Mrs.  G's  tax,  who  has  property  to  the  amount  of 
$  7980  ?  Ans.  $  23.94. 

5.  If  H  pays  for  2  polls,  and  has  property  to  the  amount  of 
$  4790,  what  is  his  tax  ?  Ans.  $  17.37. 

6.  M's  real  estate  is  valued  at  $  9280,  and  his  personal  prop- 
erty at  $  3600  ;  what  is  his  tax,  if  he  pays  for  4  polls  ? 

Ans.  $  44.64. 


EQUATION    OF    PAYMENTS. 

229.  Equation  of  PajmentS  is  the  process  of  finding  the  average 
or  mean  time  when  the  payment  of  several  sums,  due  at  different 
times,  may  all  be  made  at  one  time,  without  loss  either  to  the 
debtor  or  creditor. 

2S0.    When  the  several  sums  have  the  same  date. 

Ex.  1.  John  Jones  owes  Samuel  Gray  S  100';  $  20  of  which 
is  to  be  paid  in  2  months,  S  40  m  6  months,  $  30  in  8  months, 
and  $10  in  12  months;  what  is  the  avei-age  time  for  the  pay- 
ment of  the  whole  sum  ?  Ans.  6mo.  12da. 

oPKRATioN.  Tlie  interest  of  S  20  for  2  mo. 

20X2=40  is  the  same  as  the  interest  of  S  1 

4  0  X      6  =  240  .    for  40  mo.  ;  and  of  S  40  for   6 

3Qv/      8=240  ^'^•i  ^'^^  same  as  of  S  1  for  240 

ins/19 190  ^^'^' '  ^^^  °^  ^  ^^  ^^^  ^  ™^*'  *^® 

■^"^  same  as  of  $  1  for  240  mo. ;  and 


10  0       1  0  0  )  6  4  0  (  6  mo.  of  S  10  for  1 2  mo.,  the  same  as 

6  0  0  of  $  1   for  1 20  mo.     Hence,  the 

interest  of  all  the  sums  to  the 

4  0  time  of  payment,  is  the  same  as 

3  0  the  interest  of  S  1  for  40  +  240 


1  0  0  ^  TTol)  r  1  2  di        4-  240  -f  1 20  =  640  mo.     Now, 

1  o  n  n  J^'  ^  1  ^•t'qulre  640  mo.  to  gain  a 

^  ^'^^  certain  sum,  S  20  +  S  40  -fS  30 

-[-  §  10  =  S  100  will  require  ^  J  y 


220.  Wliat  is  equntion  of  payments  ?  — 230.  Why  in  tlie  example  do  we 
multiply  the  $  20  Ijy  2  ? 


EQUATION   OF   PAYMENTS.  231 

of  640  mo. ;  and  640  mo.  -|-  100  =  6  mo.  12  da.,  the  average  or  mean 
time  for  the  payment  of  the  whole.     Hence  the 

KuLE.  —  Multiply  each  payment  by  its  own  time  of  credit,  and  divide 
the  xuin  of  the  products  by  the  man  of  the  payments. 

Note  1.  —  This  is  the  rule  usually  adopted  by  merchants,  but  it  is  not 
perfeclli/  correct;  for  if  1  owe  a  man  $iiOO,  $100  of  which  I  am  to  pay 
down,  and  the  other  $  100  in  two  years,  the  equated  time  for  the  pa3'mcnt 
of  both  sums  would  be  one  year.  It  is  evident  that,  for  dcfenintr  tl'.c  pay- 
ment of  the  first  $  100  for  1  year,  1  ought  to  pay  the  amount  of  $  100  for 
that  time,  which  is  $106;  but  for  the  otlicr  $100,  which  I  pay  a  year 
before  it  is  due,  I  oujjht  to  pay  the  present  worth  of  $  100,  which  is  $94..33f^ 
and  $106  +  $94.3.3fi-=  $200.33|J-;  whereas,  by  the  mercantile  method 
of  equating  payments,  I  only  pay  $  200. 

Note  2.  —  When  a  payment  is  to  be  made  dorm  it  has  no  product,  but  it 
must  be  added  with  the  other  payments  in  finding  the  average  time. 

Examples  for  Practice. 

2.  John  Smith  owes  a  merchant  in  Boston  $  1000,  $  250  of 
which  is  to  be  paid  in  4  months,  $  350  in  8  months,  and  the  re- 
mainder in  12  months ;  what  is  the  average  time  for  the  payment 
of  the  whole  smn?  Ans.  8mo.  18da. 

3.  A  gentleman  purchased  a  house  and  lot  for  $1560,  ^  of 
which  is  to  be  paid  in  3  months,  -5^  in  G  months,  ^  in  8  months, 
and  the  remainder  in  10  months;  what  is  the  average  titne  of 
payment  ?  Ans.  7y^g'^  months. 

4.  Samuel  Church  sold  a  farm  for  $  4000  ;  $  1000  of  which  is 
to  be  paid  down,  $  1000  in  one  year,  and  the  remainder  in  2 
years;  but  he  afterwards  agreed  to  take  0  note  for  the  whole 
amount ;  for  what  time  must  the  note  be  given  ? 

Ans.  15  months. 

5.  A  wholesale  merchant  in  Boston  sold  a  bill  of  merchandise 
to  the  amount  of  S  5000  to  a  retail  merchant  of  Exeter,  N.  H. ; 
he  is  to  pay  ^  of  the  money  down,  -J  of  the  remainder  in  6  months, 
f  of  what  tlien  remains  in  9  months,  and  the  rest  at  the  end  of 
the  year.  If  he  wishes  to  pay  the  whole  at  once,  what  wiil  be 
the  average  time  of  payment  ?  Ans.  6mo.  27da. 

231.    When  the  several  sums  have  different  dates. 

Ex.  1.  Purchased  of  James  Brown,  at  sundry  times,  and  on 

231.  The  rule  for  equation  of  payments  ?  Is  the  rule  perfectly  correct  1 
Explain  wliy  it  is  not.  When  a  payment  is  to  be  Mjade  down,  what  is  to  be 
done  with  it  ? 


232  EQUATION  OF  PAYMENTS. 

various  terms  of  credit,  as  by  the  statement  annexed.     When  is 
the  medium  time  of  payment  ? 


Jan.          1, 

a  bill  amounting  to 

$  360,  on  3  months'  credit. 

Jan.       15, 

do.             do. 

186,  on  4  months'  credit. 

March     1, 

do.             do. 

450,  ou  4  months'  credit. 

May      15, 

do.             do. 

300,  on  3  months'  credit. 

June      20, 

do.            do 

500,  on  5  months'  credit. 
Ans.  July  25,  or  in  115  da. 

Due  April 
May 
July 
Aug. 

OPERATION. 

1,    3  60 
15,    186X     44=         8184 

1,    450X     91=      40950 
15,    300x136=      40800 

Nov. 

20,    5  0  0X233 

=   1  1  6500 

1796 

)  2  0  6  4  3  4  (  1  1  4|f  1  days. 
1796 

2683 
1796 

8874 
7  184 

1690 

We  first  find  the  time  when  each  of  the  bills  will  become  due. 
Then,  since  it  will  shorten  the  operation,  loe  take  the  first  time  when  any 
bill  becomes  due,  instead  of  its  date,  for  the  period  from  which  to  com- 
pute the  average  time.  Now,  since  April  1  is  the  period  from  which 
the  average  time  is  computed,  no  time  will  be  reckoned  on  the  first 
bill,  but  the  time  for  the  payment  of  the  second  bill  extends  44  days 
beyond  April  1,  and  we  multiply  it  by  44.  (Art.  230.)  Proceeding 
in  like  manner  with  the  remaining  bills,  we  find  the  average  time  of 
payment  to  be  115  days  nearly,  from  April  1,  or  on  the  25th  of  July. 
Hence, 

Find  the  time  when  each  of  the  sums  becomes  due.  Multiply  each  sum 
by  the  number  of  days  intervening  between  the  date  of  its  becoming  due 
and  the  earliest  date  on  which  any  sum  becomes  due.  Then  proceed  as 
in  the  rule  (Art.  230),  and  the  quotient  will  be  the  average  time  required, 
in  days  forward,  from  the  date  of  the  earliest  su77i  becoming  due. 

Note.  — In  the,  work,  if  there  be  a  fraction  of  a  day  less  than  i,  it  may 
be  rojectcd  ;  but  if  },,  or  more  than  ^,  it  may  he  reckoned  as  I  day. 

2.31.  Tho  rule  for  finding  the  .ivcrago  tim(>,  when  thore  ai-o  different  dates? 
By  what  other  method  can  you  olnain  nearly  the  same  result  ? 


u 

16, 

u 

do. 

do. 

Feb. 

11, 

(( 

do. 

do. 

u 

23, 

(( 

do. 

do. 

Mar. 

19, 

i( 

do. 

do. 

EQUATION   OF   PAYMENTS.  233 

Examples  for  Practice. 

2.  I  have  purchased  several  parcels  of  goods,  at  sundry  times, 
and  on  various  terms  of  credit,  as  by  the  following  statement. 
"What  is  the  average  time  for  the  payment  of  the  whole  ? 

Jan.      1,  1856,  a  bill  amounting  to  $  175.80,  on   4  months'  cr. 

96.46,  on  90  days' 
78.39,  on    3  months'   " 
49.63,  on  60  days' 
114.92,  on     6  months'   « 
Ans.  May  30,  or  in  45  da. 

3.  Sold  S.  Dana  several  parcels  of  goods,  at  sundry  times,  and 
on  vai'ious  terms  of  credit,  as  by  the  following  statement : 

Jan.      7,  1854,  a  bill  amounting  to  $  375.60,  on  4  months'  cr. 
April  18,       "         do.         do.  687.25,  on  4  months'    " 

June     7,       "         do.         do.  568.50,  on  6  months'    " 

Sept.  25,       "         do.         do.  300.00,  on  6  months'    " 

Nov.     5,       "         do.         do.  675.75,  on  9  months'    " 

Dec.      1,       "         do.         do.  100.00,  on  3  months'    " 

What  is  the  average  time  for  the  paj^Tnent  of  all  the  bills  ? 

Ans.  Dec.  24,  or  in  231  da. 

4.  The  following  is  my  account  against  G.  M.  Holbrook,  and 
I  wish  to  ascertain  the  average  time  of  payment. 

Jan.     1,  1857,  97  yards  of  broadcloth,  at  $  4.50,  on    3  mos.'  or. 

7  bales  of  cotton  cloth,  "    18.50,  on  60  days'    " 
9  tons  of  iron,  «   45.00,  on    4  mos.'    " 

11  hhds.  of  molasses,      «    12.00,  on  30  days'   " 

8  doz.  shovels,  «     9.00,  on    2  mos.'   " 
14cwt.  of  sugar,               "      6.50,  on     1  mo.'s    " 

8  chests  of  tea,  "    15.00,  on  90  days'   " 

Ans.  July  16,  or  in  106  da. 

5.  The  followmg  is  an  account  of  my  bills  against  J.  Crowell ; 

Jan.    1,    1854,  a  bill  amounting  to  $  300,  on  6  months' credit. 

do. 
do. 
do. 
.do. 
do. 
do. 

"What  is  the  average  time  of  payment  on  the  above  bills  ? 

Ans.  March  9,  1856,  or  in  20  mo.  8  da. 
20* 


Feb.  10, 

li 

May    1, 
June  15, 

ii 

July     5„ 
Sept.  25, 
Dec.    1, 

ii 
ii 
ii 

June 

ii 

do. 

Sept. 

a 

do. 

Feb. 

1855, 

do. 

July 

1856, 

do. 

Dec. 

a 

do. 

May 

1857, 

do. 

500, 

on 

5 

months' 

li 

200, 

on 

6 

months' 

ii 

800, 

on 

8 

months' 

ii 

400, 

on 

9 

months' 

ii 

900, 
100. 

on 

nn 

7 

months' 

ii 
i( 

234  EQUATION   OF   PAYMENTS. 

232.  When  partial  payments  have  been  made  before 
the  debt  is  due. 

Ex.  1.  I  have  purchased  goods  to  the  amount  of  $  800,  on  a 
credit  of  6  months.  At  the  end  of  2  months  I  pay  $  100,  and  at 
the  end  of  another  month  I  pay  $  200  more.  How  long;,  in 
equity,  after  the  expix'ation  of  the  6  mouths,  ought  the  balance 
to  remain  unpaid  ?  '        Ans.  2  months. 

OPERATION.  The   interest   on    tlie 

100X4=,  400  $100   for   4   months   is 

200X3=  600  equal  to  the  interest  of 

S  1  for  400  mouths ;  and 

•  300        500)1000  the  interest  of  the  S  200 

~"  for  3  months,  to  that  of 

$8  0  0  — $3  0  0  =  $5  0  0.  '^"^'  SI  for  600  months;  and 

thus  the  interest  on  both 
partial  payments,  at  the  expiration  of  the  6  mouths,  is  equal  to  the 
interest  of  $  1  for  400  -|-  GOO,  or  1000  months.  To  equal  this  credit 
of  interest,  the  balance  of  the  debt,  which  we  find  to  be  $  500,  should 
remain  unpaid,  after  the  6  months,  -gl^  of  1000  months,  or  2  months. 

Rule.  —  Multiply  each  payment  by  the  time,  in  months  or  days,  it 
was  made  before  it  became  due,  and  divide  the  sum  of  the  products  by 
the  balance  remaining  unpaid.     2'he  quotient  will  be  the  required  time. 


Examples  for  Practice.  • 

2.  Sold,  March  11,  1855,  James  Stone  goods  to  the  amount  of 
$  1850,  on  a  credit  of  4  months.  I  received  from  him,  April  7, 
$400;  May  15,  $270;  and  June  20,  $350.  When  in  equity 
should  I  receive  the  balance  ?  Ans.  Sept.  22,  1855. 

3.  Bought,  June  12,  1855,  of  William  Jones,  goods  to  the 
amount  of  $  1200,  on  a  credit  of  8  months.  I  paid  him,  Septem- 
ber 1,  $400;  November  1,  $200;  and  December  1,  $100. 
When  in  equity  can  he  require  the  balance  of  me  ? 

Ans.  Aug.  17,  1856. 

4.  I  sold,  September  25,  1855,  John  Eckles  144  barrels  of 
flour,  at  $  12  per  barrel,  and  370  bushels  of  wheat,  at  $3  per 
busliel,  on  6  months'  credit.  I  received  of  him,  September  25, 
S  1000  ;  November  1,  $800  ;  and  December  21.  $  600.  When 
ought  I  to  be  paid  the  remainder?  Ans.  June  14,  1858. 

2.^2.  The  nilc  for  fiiuhncr  the  iivernL'e  lime  of  paying  tlie  balance  of  a 
debt,  when  partial  payments  have  been  made  1 


EQUATION   OF   PAYMENTS. 


235 


5.  Wilson  vSeymour  boiiglit  March  20,  1855,  of  a  merchant  in 
Troy,  merchandise  to  the  amount  of  $2000,  on  6  months'  credit. 
lie  paid  down  $  500  ;  May  10,  $350,  and  June  7,  %  400.  AVhen 
did  the  balance  become  due  ?  Ans.  May  18,  1856. 

233t  When  an  account  containing  items  of  both  debit 
and  credit. 

Ex.  iT  At  wliat  time  did  the  balance  of  the  following  account 
become  due,  allowing  that  each  item  drew  interest  from  its  date  ? 


Dr. 


Martin  Jordan  in  account  with  David  HiU  S,-  Co.  Or. 


1856. 

1 

j      1856. 

Jan.  22, 

To  merchandise, 

$89loO 

i   Jan.     4, 

By  merchandise. 

$77  00 

"     24, 

7600 

Apr.  16, 

«                      K 

40  00 

Feb;  20, 

25  00 

May  14, 

tt           « 

143  00 

"     23, 

210  00 

April  4, 

189  00 

May  21, 

30  00 

Ans.  February  9,  1856. 


Jan.  22. 

"  24, 
Feb.  20, 

"  23, 
April  4, 
May  21, 


Debits. 

89 

76X  2  =     152 

25x  29=  725 
210X  32=  6720 
189X   73  =  13797 

30X120=  3600 


OPERATION. 


619 


days. 
)24994(40|3.t 

2476 


234 


Credits. 

Jan.      4,     77 

April  16,    40X103=  4120 

May   14,  143x131  =  18733 


2G0 


dars. 
)22853(87-yf 
2080 


2053 
1820 

233 


Average  date  of  purchase,  40  days 
from  Jan.  22,  or  on  March  2. 

Difference  between  March  2 
and  April  1=30  days, 


Average  of  credits,  88  days  from 
Jan.  4,  or  on  April  1. 

260 
30 


$619  —  $260=$  359,  or  balance, 


359)7800(2l||ldays. 
718 


620 
359 


261 


22  days  back  from  March  2=  February  9. 


233.  How  do  you  equate  a-n  account  having  items  of  debit  and  credit  ? 


236 


EQUATION  OF  PAYMENTS. 


On  equating  each  side  of  the  account  (Art.  230),  we  find  the  debits 
$Cli),  became  due  40  days  from  January  ■J-2,  or  on  March  2  ;  and  the 
credits,  ;$  2(J0,  became  due  ii6  days  tiom  January  4,  or  ou  April  1. 

If  the  account  had  been  settled  on  March  2,  it  is  evident  the  "credits, 
S  260,  -would  have  been  paid  30  days,  or  the  time  from  March  2  to 
April  1,  bejbre  having  become  due.  This  would  have  been  a  loss  of 
the  use  or  interest  of  that  sum  to  the  credit  side  of  the  account,  and  a 
corresponding  gain  to  the  debit  side.  Now,  as  the  settlement  is  re- 
quired to  be  one  of  equity,  we  find  how  long  it  will  take  the  balance 
of  the  account,  $  359,  to  gain  the  same  interest  that  S  260  would  gain 
in  the  30  days.  If  it  takes  $  260  to  gain  a  certain  interest  in  30  days, 
it  would  take  S  1  to  gain  the  same  interest  260  times  30  days,  or  7800 
days  ;  and  S  359  to  gain  the  same  amount  of  interest  -g\^  of  7800  days, 
or  22  days  nearly.  Hence,  the  balance  became  due  22  days  back" of 
March  2,  or  on  February  9. 

In  this  example,  the  time  was  counted  back  from  the  average  date 
of  the  larger  amount,  since  it  became  due_fir.it ;  but  when  that  amount 
becomes  due  last,  the  time  is  counted  forward  from  its  average  time. 

Rule.  —  Find  the  average  time  of  each  side  becoming  due. 

Mulliphj  the  amount  of  the  smaller  side  h>j  the  number  of  days  between 
the  two  aoerage  dales,  and  divide  the  product  hi/  the  balance  of  the  ac- 
count. 

The  quotient  will  he  the  time  of  the  balance  becoming  due,  counted 
from  the  average  dale  of  the  larger  side,  r..\CK  when  the  amount  of  that 
side  is  due  first,  but  fouward  when  it  is  due  last. 

Note.  —  Having  the  average  time  of  a  balance  becoming  due,  its  cash 
VALUB  can  be  ascortained  iclien  the  bdl'tiire  is  due  before  the  time  of  sdtlimj 
the  account,  hij  addinrj  to  it  the  interest  up  to  the  time  of  settlement,  and  irlien  due 
after  that  time,  hi/  fnding  the  present  icorlh  (Art.  213)  feom  the  time  of  stttle- 
vieni  to  the  time  of  the  balance  becoming  due. 

Examples  for  Practice. 

2.  In  settling  the  following  account,  when  did  the  balance  be- 
come due,  the  merchandise  items  being  on  G  monllis'  credit  ? 


Dr.  Jllram  Lewis  in  account  with  Joseph  Wan-en. 


Cr. 


1854. 

1854. 

1 

Feb.  16, 

To 

racrchandise, 

$37.5 

80 

Mar.  20, 

By  cash, 

$300  00 

April  8, 

If 

<i 

432 

18 

Juno  17, 

"   mcrcliandise, 

371  .50 

Mav  17, 

tt 

a 

320 

1.5 

July     4, 

"   cash. 

200  00 

Jufv  13, 

it 

tt 

L-isiia  1 

Sept.  25, 

"   niercliandise, 

85  20 

Ans.  March  8,  1 855. 


.323.  What  is  the  rule? 
account  bo  found  .' 


IIow  can  tlie  ca.sli  value  of  the  balance  of  an 


RATIO.  237 

3.  Edward  Doton  owes  Daniel  Stetson,  1855,  May  1,  for 
merchandise,  §  5U0  ;  May  15,  for  tinibei',  $400  ;  June  14,  for  a 
horse,  $  300  ;  July  24,  tor  bill  of  labor,  $  100.  Stetson  owes 
Doton,  1855,  March  7,  for  a  pleasure-boat,  $400;  April  2,  for 
merchandise,  $200;  May  G,  for  merchandise,  $300;  June  13, 
for  a  carriage,  $  120.  Allowing  all  the  items  to  be  on  6  months' 
credit,  when  will  the  balance  of  the  account  become  due  ? 

Ans.  April  27,  1856. 


RATIO 


1 


234i  Ratio  is  the  relation,  in  respect  to  magnitude  or  value, 
which  one  quantity  or  number  has  to  another  of  the  same  kind, 
or  the  quotient  arising  from  the  division  of  one  number  by  an- 
other.    Thus,  the  ratio  of  6  to  3  is  2. 

Of  the  two  numbers  necessary  to  form  a  ratio,  the  first  is  called 
the  Antet'Cdcnt,  and  the  last  the  ConseqUCllt.  Thus,  in  the  example 
given,  6  is  the  antecedent,  and  3  the  consequent. 

A  Simple  Ratio  is  that  having  but  one  antecedent  and  one  con- 
sequent. 

The  Terms  of  a  ratio  are  its  antecedent  and  consequent. 

23-5.  A  ratio  may  be  expressed  in  two  ways.  The  ratio  of  6 
to  3  may  be  expressed  by  two  dots  ( : )  between  the  terms  ;  thus, 
G  :  3  ;  or  in  the  form  of  a  fraction,  by  making  the  antecedent  the 
numerator  and  the  consequent  the  denominator,  thus,  f. 

The  terms  of  a  ratio  must  be  of  the  satne  hind,  or  such  as  may 
be  reduced  to  the  same  denomination!  Thus,  shillings  have  a 
ratio  to  shillings ;  but  sliillings  have  not  a  ratio  to  gallons,  nor 
pounds  to  days. 

236.    A  ratio  may  be  either  direct  or  inverse. 

A  Direct  Ratio  is  when  the  antecedent  is  divided  by  the  conse- 
quent. 

An  Inverse  Ratio  is  when  the  consequent  is  divided  by  the  ante- 
cedent. Thus,  the  direct  ratio  of  6  to  3  is  f ,  and  the  inverse 
ratio  of  6  to  3  is  f,  or  ^. 

234.  "What  is  ratio?  How  many  numbers  are  necessary  to  form  a  ratio? 
What  are  the  antecedent  and  consequent  called  ?  —  235.  What  two  ways  are 
there  of  expressing  a  ratio?  —  236.  What  is  a  direct  ratio?  An  inverse 
ratio  ? 


238  RATIO 

77?e  DIRECT  ratio  of  one  number  to  another  is  found  hy  divid' 
ing  the  number  whose  ratio  is  required,  which  is  the  antecedent,  by 
the  mimber  with  wliich  it  is  compared,  wldch  is  the  consequent. 

The  INVERSE  ratio  is  found  by  reversing  this  process. 


Examples  for  Practice. 

1.  What  is  the  direct  ratio  of  9  to  3  ?  An?.  3.  Of  18  to  G  ? 
Of  16  to  4?     Of  24  to  12?     Of  20  to  5  ?     Of  15  to  3? 

2.  What  is  the  direct  ratio  of  7  to  21  ?  Ans.  |.  Of  4  to 
28?     Of  6  to  30?     Of  9  to  11?     Of  9  to  99  ?     Of  30  to  90? 

3.  What  is  the  direct  ratio  of  60  to  12?  Of  132  to  11  ? 
Of  40  to  120?     Of  32   to  96?     Of  200   to  50?     Of  144  to 

1728  ?     Of  300  to  GO  ? 

4.  What  is  the  inverse  ratio  of  10  to  5  ?  Ans.  J-.  Of  27  to 
81  ?     Of  16  to  48  ?     Of  72  to  9  ?     Of  11  to  88  ? 

5.  What  is  the  direct  ratio  of  2£  os.  to  9s.  ?  Ans.  5.  Of  9in. 
to  1ft.  6  in.  ? 

237.  A  Componiul  Ralio  is  the  product  of  two  or  more  ratios. 
Thus  the  ratio  compounded  of  the  ratios  of  8 : 4  and  12:3  is 

1  X  V-  =  ff  =  8,  or  8  X  12  :  4  X  3  =8. 

A  compound  ratio  is  generally  expressed  by  writing  the  ratios 

8:4 
of  which  it  is  composed,  one  above  the  other.     Thus,  -i  9  .  o  ex- 
presses a  compound  ratioo 

One  quantity  is  said  to  vary  directly  as  another,  when  both 
increase  or  decrease  together  in  the  same  ratio. 

One  quantity  is  said  to  vary  inversely  as  anotlier,  when  the  one 
increases  in  the  same  ratio  as  the  other  decreaseso 

If  the  terms  of  a  ratio  are  both  midfiplicd  or  divided  by  the 
same  number,  the  ratio  is  not  altered.  Thus,  the  ratio  of  S  '.'Z 
is  4 ;  the  ratio  8x2:2X2  is  4;  and  the  ratio  of  8  -7-  2  :  2  -^ 

2  is  4. 


237.  What  is  a  compound  ratio?  "What  a  duplirate  ratio?  Wliat  a 
triplicate  ratio?  What  is  the  cflFcet  of  inuhiplying  or  dividing  the  terms  of 
a  ratio '' 


PROPOKTION.  239 


PROPORTION. 

238.  Proportion  is  an  equality  of  ratios.  Tlius  9  :  3  =  12  :  4 
expresses  a  proportion. 

Proportion  is  usually  written  with  four  dots  (::),  instead  of  the 
sign  of  equality  between  the  ratios  ;  thus,  9  :  3  : :  12  :  4  expresses 
a  proportion,  and  is  read,  the  ratio  of  9  to  3  is  equal  to  tlie  ratio 
of  12  to  4,  or  9  is  to  3  as  12  is  to  4. 

The  numbers  which  foi-m  a  proportion  are  called  Proportionals.. 
The  frst  and  third  are  called  Antecedents,  the  second  and  fourth 

are  called  Consequents ;  also,  the  jirst  and  last  are  called  Extremes, 

and  the  remainiuo;  two  the  JUcans. 


o 


239o  Any  four  numbers  are  said  to  be  proportional  to  each 
other  when  the  first  contains  the  second  as  many  times  as  the 
third  contains  the  fourth ;  or  when  the  second  contains  the  first 
as  many  times  as  the  fourth  contains  the  third.  Thus,  9  has  the 
same  ratio  to  3  that  12  has  to  4,  because  9  contains  3  as  many 
times  as  12  contains  4. 

240.  In  a  proportion,  if  the  antecedents  or  consequents,  or  hoth, 
are  divided  hy  the  same  number,  they  are  stiU  proportionals.  Thus, 
dividing  the  antecedents  of  the  proportion  4 :  8  : :  10  :  20  by  2,  we 
have  2  :  8  : :  5  :  20  ;  dividing  the  consequents  by  2,  w£  have 
4  :  4  : :  10  :  10  ;  and  dividing  both  the  consequents  and  antece- 
dents by  2,  we  have  2  :  4  : :  5  :  10  ;  each  of  which  is  a  proportion, 
since  if  we  divide  the  second  term  of  each  by  the  fii'st,  and  the 
fourth  by  the  tliird,  the  two  quotients  will  be  equal.  The  effect 
is  the  same  when  the  terms  are  multiplied  by  the  same  number. 

241.  In  a  proportion,  the  product  of  the  extremes  is  equal  to  the 
product  of  the  means.  Thus,  the  proportion,  14  :  7  : :  18  :  9  may 
be  expressed  fractionally,  J/-  =  J^.  Now,  if  we  reduce  these 
fractions  to  a  common  denominator,  we  have  ^i^-P-  =  J^H#- ;  but  in 
this  operation  we  multiplied  together  the  two  extremes  of  the  pro- 
portion, 14  and  9,  and  the  two  means,  18  and  7  ;  thus,  14  X  9 
=  18  X  7. 

238.  What  is  proportion  ?  How  is  proportion  written  ?  Wliat  are  the 
numbers  called  that  form  a  proportion  ?  Which  are  the  extremes  ?  Which 
the  means  ?  —  239.  When  are  numbers  said  to  be  in  proportion  to  each 
other?  —  240.  What  is  tlic  effect  of  dividing  tlie  ant<^cedents  or  conscquent.s 
of  a  proportion?  Of  multiplyin<j  them  ?  —  241.  How  does  the  product  of 
the  extremes  compare  with  that  of  the  means  ? 


240  SIMPLE   PROPORTION 

242i  If  the  extremes  and  one  of  the  means  are  given,  the  other 
mean  may  be  found  by  dividing  the  product  of  the  extremes  by 
the  given  mean.  Tliu>,  if  the  extremes  are  3  and  24,  and  the 
given  mean  6,  the  other  mean  is  12  ;  because  24  X  3  ^  72  ; 
and  72  ^  6  =  12= 

2i3o  If  the  means  and  one  of  the  extremes  are  given,  the  other 
extreme  may  be  found  by  dividing  the  product  of  the  means  by  the 
given  extreme.  Thus,  it"  the  means  are  8  and  16,  and  the  given 
extreme  4,  the  other  extreme  is  32;  because  16  X  8  =  128; 
and  128  -r-  4  =  32. 

SIMPLE  PROPORTION. 

224o  Simple  Proportion  is  an  equality  between  two  simple 
ratios. 

Simple  Proportion  is  sometimes  called  the  Rule  of  Three,  from  three 
terms  beinfr  given  to  find  a  fourth. 


-"tJ  o' 


OPERATION. 

treme. 

Jlean.               Mean. 

71b. 

:  3  6  lb. :  :  5  6  cts. 

36 

245.    To  state  and  solve  questions  in  Simple  Propor- 
tion. 

Ex.  1.  If  71b.  of  sugar  cost  56  cents,  Avhat  will  361b.  cost  ? 

Ans.  $.2.88. 

Sin-ce  7lb.  have  the  same  ratio  to 

SGlb.   as  56   cents,  the   cost  of  the 

former,  have  to  the  cost  of  the  latter, 

we  have  the  fii'st  three  terms  of  a 

proportion  given,  namely,  one  of  the 

3  3  6  extremes  and  the  two  means.     Xow, 

16  8  to  arrange  the  given  nimibers  in  the 

order  of  a   proportion,  or  stale   the 

7  )  2  0.1  6  question,  we  make  50  cents  the  third 

$o  o  Q  T,  .  term,  because  it  is  of  tlie  same  kind, 

Z.O  o  Extreme.  ,  ,'       .,  .•    .     ^i  •      i 

and  has  the  same  ratio  to  the  I'cqun-ca 

answer,  or  fourth  term,  as  the  first  has  to  the  second.     From  the  nature 

of  the  question,  since  the  answer  will  be  more  than  56  cents,  or  the 

third  term,  the  second  term  must  be  larger  than  the //>>■?/;  Ave  make  the 

36  the  second  term,  and  the  7  the  first,  and  then  the  product  of  the 

means  divided  by  the  given  extreme,  gives  the  required  exti-cme. 

(Art.  243.) 


242.  If  the  extremes  and  one  of  the  means  are  pivon,  how  can  the  otlicr 
mean  be  found?  —  243.  When  the  moans  and  one  of  the  extremes  are  given, 
how  can  the  other  extreme  he  fuinid  ?  —  244.  Wliat  is  simple  proportion? 
How  many  terms  are  given  in  questions  in  simple  proportiou  ? 


SIMPLE   PROPORTION.  241 

By  Analysis.  —  If  7lb.  cost  56  cents,  1  pound  will  cost  \  of  56 
cents,  or  8  cents.  Then,  if  lib.  cost  8  cents,  3Glb.  will  cost  36  times  8 
cents,  or  $  2.88. 

Ex.  2.  If  76  barrels  of  flour  cost  $  456,  what  will  12  barrels 
cost?  Ans.  $  72. 

OPERATION.  We  state  this  question  by  making 

bar.    bar.             S  $  456  the  third  term,  because  it  is  of 

7  6:12:  :  4  5  6  tlie  same  kind  of  the  required  term. 

1  2  Then,  since  the  answer  must  be  less 

_„..,„.^,H,__  than  $456,  because   12  barrels  will 

7b)04:i'Z{^ifilZ  pQg|.  ]ggg  j.|j^^j^  r.^  barrels,  we  make  12 

*^  3  2  barrels,  the  smaller  of  the  two  terms, 

the  second  term,  and  76  barrels  the 
Jirst  term,  and  proceed  as  before. 


By  Analysis.  — If  76  barrels  cost  $456,  1  barrel  will  cost  -^  of 
S456,  or  $  6.  Then,  if  1  barrel  cost  $  6,  12  barrels  will  cost  12  times 
$  6,  or  $  72, 

Ex.  3,  If  3  men  can  dig  a  well  in  20  days,  how  long  will  it 
take  12  men?  Ans.  5  days. 

OPERATION.  Since  the  required  answer  is  days,  we  make 

men.    men.    days.  20  days  the  third  term.     And  as  12  men  will 

1  2  :  3  :  :  20  dig  the  well  in  less  time  than   3  men,  the 

3  answer  must  be  less  than  20  days.     There- 

—  fore  we  make  the  3  the  second  term,  and  the 

12)  6^  12  the  first,  and  proceed  as  in  the  other  ex- 

5  day?.      ^^P^«^- 

By  Analysis.  —  If  3  men  can  dig  the  well  in  20  days,  it  will  take 
one  man  3  times  20  days,  or  60  days.  Again,  If  one  man  can  dig  the 
well  in  60  days,  12  men  can  dig  it  in  ^  of  60  days,  or  5  days. 

Rule.  —  Write  for  the  tMrd  term  that  number  which  is  of  the  same 
kind  as  the  required  fourth  term. 

Of  the  other  two  numbers,  write  the  larger  for  the  second  term,  and  the 
less  for  the  first,  when  the  answer  should  exceed  the  third  term;  but  lorite 
the  less  for  the  second  term,  and  the  larger  for  the  first,  when  the  answer 
shofild  be  less  than  the  third  term. 

Multiply  the  second  and  third  terms  together,  and  divide  their  product 
by  the  first. 

245.  What  is  meant  by  stating  the  question  ?  "Which  of  the  terms  given 
in  the  example  do  you  make  the  third  1  Why "?  Which  the  second  ?  Why  ? 
Which  the  first  ?  Why  1  After  the  question  is  stated,  how  do  you  obtaia 
the  answer  ? 

21 


242 


SDIPLi:   rROPOETION. 


Note  1.  —  Wlien  the  first  and  second  terms  consist  of  different  denomi. 
nations,  tliey  must  be  rcdtu-ed  to  the  same  denomination  ;  and  when  the 
tliird  term  is  a  compound  number,  it  must  be  rcduecd  to  the  lowest  denomi- 
nation mentioned  in  it.  The  answer  will  be  the  same  denomination  as  the 
third  term. 

Note  2.  —  To  shorten  the  operations,  factors  common  to  tlie  dividend 
and  divisor  may  be  cancelled. 

Note  3.  — It  is  often  a  convenient  method,  to  divide  the  third  term  hy  the 
ratio  of  thejirst  term  to  the  second. 

Ex.  4.   If  16  bushels  of  wheat  are  worth  $24,  what  are  96 
bushels  worth  ?  Ans.  $144. 


operation  by  cancellation. 


bu. 
16 


bu. 

96 
6 

$24X00 


24 


=  $144 


We  first  state  the  question  as  di- 
rected in  the  rule,  and  then  write 
the  second  and  thiixi  terms  above 
a  horizontal  line,  with  the  sign  of 
multiplication  between  them,  for  a 
tUvidond,  and  the  first  term  below 
the  line  for  a  divisor,  and  cancel  the 
common  factors. 

Again,  we  place  the  S  24,  which 
is  of  the  same  kind  of  the  required 
answer,  above  a  line  for  a  dividend  ; 
and  then  say.  Since  16  bushels  arc 
worth  S  24,  1  bushel  is  worth  Jg  of 
$  24,  and  express  the  division  by 
placing  the  16  below  the  line  lor  a 
since   96   bushels  are  worth   96   times  as  much  as  1 

bushel,  we  express  the  multiplication  by  placing  the  96  above  the  line, 

and  then  cancel  as  before. 


BY  ANALYSIS  AND  CANCELLATION. 
6 


$24X00 


=  $144 


^0 


divisor. 


EXASIPLES   FOE   PRACTICE. 

5.  What  cost  9  gallons  of  molasses,  if  63  gallons  cost  $  H.49  ? 

Aus.  $  2.01. 

6.  What  cost  ST-aeres-ef-laud^  if  19  acres  can  be  obtained  for 
$337.25?         '  '      ~^         Ans.  $1721.75. 

7.  If  a  man  travel  319  miles  in  11  days,  how  far  will  lie  travel 
in  47  days?  Ans.  1363  miles. 

8.  AVlicn  $  120  are  paid  for  15  barrels  of  mackerel,  wliat  will 
be  the  cost  of  79  barrels?  Ans.  $  632. 


24.5.  "Wliat  is  the  mlc  for  simple  proportion  ?     IIow  should  tlie  pupil  ^lor- 
form  the  questions  ?     How  do  you  state  the  (piestion  and  arranj^o  tlie  toniisj 
for  ranceliation '?     "What  do  you  cancel  !     IIow  do  you  arrange  the  terms 
for  cancellation  by  analysis  ? 


1 


SIMl'LE   rROPORTION.  243 

9.  If  9  horses  eat  a  load  of  hay  in  12  days,  how  many  horses 
can  eat.  the  same  quantity  in  3  days  ?  Ans.  36  horses. 

10.  When  $  5.88  are  paid  for  7  gallons  of  oil,  what  cost  27 
gallons  ?  Ans.  $  22.68. 

11.  When  $  10.80  are  paid  for  91b.  of  tea,  what  cost  1471b  ? 

Ans.  $  176.40. 

12.  Wliat  cost  27  tons  of  coal,  when  9  tons  can  be  purchased 
for  $  85.95  ?  Ans.  $  257.85. 

13.  If  15  tons  of  lead  cost  $  105,  what  cost  765  tons? 

Ans.  $  5355.00. 

14.  If  16hhd.  of  molasses  cost  $  320,  what  cost  176hhd.? 

$  3520.00. 

15.  If  15cwt.  3qr.  171b.  of  sugar  cost  $  124.67,  what  cost 
76cwt.  2qr.  191b.?  Ans.  $  600.5 6-f-. 

16.  If  the  cars  on  the  Boston  and  Maine  Railroad  go  one 
mile  in  2  minutes  and  8  seconds,  how  long  will  they  be  in  passing 
from  Haverhill  to  Boston,  the  distance  being  32  miles  ? 

Ans.  Ih.  8min.  16sec. 

17.  If  a  man  can  travel  3m.  7fur.  18rd.  in  one  houi-,  how  far 
can  he  travel  in  9h.  45min.  19sec.  ?     Ans.  38m.  2fur.  32rd.-|-. 

18.  A  fox  is  96  rods  before  a  greyhound,  and  while  the  fox  is 
running  15  rods  the  greyhound  runs  21  rods;  how  far  must  the 
dog  run  before  he  can  catch  the  fox?  Ans.  336  rods. 

19.  If  5  men  can  reap  a  field  in  12  hours,  how  long  would  it 
take  them  if  4  nienjwgi:e_ added  to  their  number  ? 

'  Ans.  6f  hours. 

20.  Ten  men  engage  to  build  a  house  in  63  days,  but  3  of 
their  number  being  taken  sick,  hcJw  long  will  it  take  the  rest  to 
complete  the  house  ?  Ans.  90  days. 

21.  If  a  4  cent  loaf  weighs  5oz.  when  flour  is  $  5  per  barrel, 
what  should  it  weigh  when  flour  is  $  7.50  per  barrel  ? 

'  Ans.  3J^oz. 

22.  If  7  men  can  mow  a  field  in  10  days  when  the  days  are 
14  hours  long,  how  long  will  it  take  the  same  men  to  mow  the 
flL'Id  when  the  days  are  13  hours  long  ?  Ans  10|5  days. 

23.  If  291b.  of  butter  will  purchase  401b.  of  cheese,  how  many 
pounds  of  butter  will  buy  791b.  of  cheese?  Ans.  57|-glb. 

24.  If  I  of  a  yard  cost  |  of  a  dollar,  what  will  -i.J  of  a  yard 
cost?  Ans.  $"0.76 /g. 

yd.     yd.          f 
3.  11.   .5.    4   V   iJ-V    "^  220  •a^OTn'^      An=; 


244  SIMPLE   PROPORTION. 

25.  If  Al  yards  of  cloth  cost  $  2 J,  what  will  19^-  yards  cost? 

Ans.  $11.50. 

yd.         yd.  $.         ci  Kici         93 

4i  :  IH  :  :  2|  ;    -^  X  "^  X  ^  =  ¥  =  ^  11-50,  Ans. 

26.  If  for  4y^Y  yards  of  velvet  there  be  received  11|  yards  ol 
calico,  how  many  yards  of  velvet  will  be  sufficient  to  purchase 
100  yards  of  calico  ?  Ans.  39|-||-  yards. 

27.  A  certain  piece  of  labor  was  to  have  been  performed  by 
144  men  in  36  days,  but  a  number  of  them  having  been  sent 
away,  the  work  was  performed  in  48  days ;  required  the  number 
of  mer^  discharged.  Ans.  36  men. 

28.  James  can  mow  a  certain  field  in  6  days,  John  can  mow 
it  in  8  days  ;  how  long  will  it  take  John  and  James  both  to  mow 
it  ?  Ans.  3^  days. 

29.  A.  Atwood  can  hoe  a  certain  field  in  10  days,  but  with 
the  assistance  of  his  son  Jerry  he  can  hoe  it  in  7  days,  and  he 
and  his  son  Jacob  can  hoe  it  in  6  days ;  how  long  would  it  take 
Jerry  and  Jacob  to  hoe  it  together  ?  Ans.  9^^  days. 

30.  Bought  a  horse  for  $  75  ;  for  what  must  I  sell  him  to  gain 
10  p*  cent.  ? 

1.00  :  1.10:  :  $75  :  $82.50,  Ans. 

31.  Bought  40  yards  of  cloth  at  $  5.00  per  yard  ;  for  what 
must  I  sell  the  whole  amount  to  gain  15  per  cent.  ? 

Ans.  $  230.00. 

32.  My  chaise  cost  $175,  but,  having  been  injured,  I  am 
willing  to  sell  it  at  a  loss  of  30  per  cent. ;  what  should  I  re- 
ceive ?  Ans.  $  122.50. 

33.  Bought  a  cargo  of  flour  on  speculation,  at  $  5.00  per  bar- 
rel, and  sold  it  at  $  6.00  per  barrel ;  what  did  I  gain  per  cent.  ? 

Ans.  20  per  cent. 

34.  Bought  a  hogshead  of  molasses  for  $  15.00,  but,  it  not 
proving  as  good  as  I  expected,  1  sell  it  for  $  12  ;  what  do  I  lose 
per  cent.  ?  Ans.  20  per  cent. 

35.  Bought  a  hogshead  of  molasses  for  $  27.50,  at  25  cents 
per  gallon  ;  how  much  did  it  contain?  Ans.  110  gallons. 

36.  A  certain  farm  was  sold  for  $  1728,  it  being  $  15.75  per 
acre;  what  was  the  quantity  of  land?     Ans.  109A.  2R.  34fp. 

37.  A  certain  cistern  has  3  cocks  ;  the  first  will  empty  it  in 
2  hours,  the  second  in  3  hours,  and  the  tliird  in  4  hours ;  in  what 
time  would  they  all  empty  the  cistern  togctlier? 

Ans.  55-^5  minutes. 


COMPOUND   PROPORTION. 


245 


COMPOUND  PROPORTION. 

246.  Compound  Proportion  is  an  expression  of  equality  between 
a  compound  and  a  simple  ratio. 

It  is  employed  in  the  solution  of  such  questions  as  require  two 
or  more  statements  in  Simple  Proportion. 

247.  To  state  and  solve  questions  in  Compound  Pro- 
portion: 

Ex.  1.  If  $100  will  gain  $8  in  12  months,  what  will  $600 
gain  in  10  months .''  Ans.  $  40. 


Extreme. 

$100 
1  2  mo. 


OPEEATION. 

Mean. 

$600 
1  0  mo. 


Mean. 

:  $8 


600  X  10  X  8     48000 


lOOX  12 


1200 


=  $40,  Extreme. 


In  stating  this 
question,  we  make 
!§  8,  the  gain,  which 
is  of  the  same  kind 
as  the  required 
term,  the  third 
term.  Then,  tak- 
inof  of  the  remain- 
mg  terms  two  of 
the  same  kind,  $  100  and  $  600,  we  inquire  if  the  answer,  depend- 
in"'  on  these  alone,  must  be  greater  or  less  than  the  third  term ; 
and  since  it  must  be  greater,  because  $  GOO  will  gain  more  than 
S  100  in  the  same  time,  we  make  $  600  the  second  term,  and  $  100 
the  first.  Again,  we  take  the  two  remaining  terms,  and  make  10  mo. 
the  second  term,  and  12  mo.  the  first,  since  the  same  sum  would 
gain  less  In  10  mo.  than  In  12  mo.  We  then  find  the  continued 
products  of  the  second  and  third  terms,  and  divide  it  by  the.  product 
of  the  first  terms,  for  the  answer. 

Rule.  —  Make  that  numher  zohich  is  of  the  same  kind  as  the  ansicer 
required  the  third  term  of  a  proportion.  Of  the  remaining  numbers,  take 
any  two,  that  are  of  the  same  kind,  and  consider  whether  an  answer, 
depending  upon  these  alone,  woidd  he  greater  or  less  than  the  third  term, 
and  place  them  as  directed  in  Simple  Proportion. 

Tlien  take  any  other  two,  and  consider  whether  an  answer,  depending 
only  upon  them,  loould  he  greater  or  less  than  the  third  term,  and  arrange 
them  accordingly ;  and  so  on  until  all  are  used. 

246.  What  is  compound  proportion  ?  For  what  is  it  employed  ?  —  247.  In 
stating  the  question,  which  of  the  numbers  do  you  make  the  third  term  ? 
Why  1  Wliat  do  you  do  witli  the  remaininaj  terms  1  How  do  you  know 
which  of  the  two  to  tnke  for  the  second  term  1  Which  for  the  first  ?  After 
all  the  terms  have  been  ai-ranged,  how  do  you  find  the  answer?  The  rule 
for  comp'ound  proportion  ? 
21* 


248 


.COMPOUND   PROPORTION. 


Multiply  the  product  of  the  second  termx  by  the  third,  and  divide  the 
reaull  by  the  product  of  the  Jirst  terms.  Tlie  quotient  will  be  the  fourth 
term,  or  answer. 

Note.  —  Operations  can  often  be  much  shortened  by  cancellation. 

Ex.  2.  If  $  100  wm  gain  $  6  in  12  months,  what  will  $800 
gain  in  8  months  ? 


OPERATION    BY    CANCELLATION. 


$100: $800 
1  2  mo.  :  8  mo 

8       4 
^00X^X0 


"}^ 


$.6 


$32 


BT  ANALYSIS  AND  CANCELLATION. 


4  8 

0  X  ^  X  ^00, 

a:^  X  -3'00 


$32 


"We  state  the  question  accord- 
ing to  the  rule,  and  then  write 
the  second  and  third  terms  for  a 
dividend  and  the  first  terms  for 
a  divisor,  and  cancel  the  com- 
mon factors. 


If  $  6  is  the  gain  of  S  100  in 
12  mo.,  in  1  mo.  the  gain  of  SlOO 
will  be  -jlj  as  much,  or  S  -^-^,  and  in 
8  mo.  8  times  as  much,  or  %  '^^. 
Again,  if  $100  gain  %^1^,^  in 
8  mo.,  S  1  will  gain  ^ig-  of  it,  or 


6X8 
'12X100 ' 


and  S800  will  gain 
800  times  as  much,  or  S  V^/vVn?°  >  and  canceUing  the  common  factors 
we  obtain  $  32  for  the  answer. 


EXA3IPLES   FOR    PRACTICE. 


3.  If  $  100  gain  $  6  in  12  months,  in  how  many  months  will 
$  800  gain  $  32  ?  Ans.  8  months. 

4.  If  $  100  gain  $  6  in  12  months,  how  large  a  sum  will  it 
require  to  gain  $  32  in  8  months  ?  Ans.  $  800. 

5.  If  $  800  gain  $  32  in  8  months,  what  is  the  per  cent.  ? 

Ans.  6  per  cent. 

6.  If  15  carpenters  can  build  a  bridge  in  60  days  when  the 
days  are  15  hours  long,  how  Idlig  will  it  take  20  men  to  build 
the  bridge  when  the  days  are  10  hours  long? 

Ans.  67^  days. 


247.  How  can  oporntions  often  be  sliortencd  ?  TTow  are  quostions  stiitotl 
for  cancellation  ?  Wliicli  tonus  are  taken  for  the  dividend  \  Which  for  tho 
divisor  ?     Wliut  are  cancelled  ? 


COMrOUND   rROPORTION.  247 

7.  If  a  regiment  of  soldiers,  consisting  of  939  men,  can  eat 
351  liushcls  of  wheat  in  3  weeks,  how  many  soldiers  will  it  re- 
quire to  eat  1404:  bushels  in  2  weeks?  Ans.  5634  soldiers. 

8.  If  8  men  spend  $  64  in  13  weeks,  what  wiU  12  men  spend 
in  52  weeks?  An.-=.  $384. 

9.  If  8  horses  consume  42  bushels  of  grain  in  24  days,  how 
many  bushels  will  suffice  32  horses  48  days  ? 

Ans.  336  bushels. 

10.  If  6  men  in  16  days  of  9  hours  each  build  a  wall  20  feet 
long,  6  feet  higli,  and  4  feet  thick,  in  how  many  daj^s  of  16  hours 
each  will  24  men  build  a  wall  200  feet  long,  16  feet  high,  and  6 
feet  thick  ?  Ans,  90  days. 

11.  If  a  man  travel  117  miles  in  15  days,  employing  only  9 
hours  a  day,  how  far  would  he  go  in  20  days,  travelling  12 
hours  a  day  ?  Ans.  208  miles. 

12.  If  12  men  in  15  days  can  build  a  wall  30  feet  long,  6  feet 
high,  and  3  feet  tljick,  when  the  days  are  12  hours  long,  in  what 
time  will  30  men  build  a  wall  300  feet  long,  8  feet  high,  and  6 
feet  thick,  when  they  work  8  hours  a  day  ?         Ans.  240  days. 

13.  If  the  carriage  of  5cwt,  3qr.  150  miles  cost  $  24.58,  what 
must  be  paid  for  the  carriage  of  7cwt.  2qr.  151b.  32  miles,  at 
the  same  rate  ?  Ans.  $  6.97-}-. 

14.  A  received  of  B  $  9  for  the  use  of  $  600  for  6  months  ; 
now  B  wishes  to  hire  of  A  $  1800  until  the  interest  shaU  amount 
to  the  same  sum.     For  how  long  must  he  hire  it  ? 

Ans.  2  months. 

15.  If  15  oxen  or  20  cows  will  eat  3  tons  of  hay  in  8  weeks, 
how  much  hay  will  be  sufficient  for  15  oxen  and  8  cows  12 
weeks  ?  Ans.  6^^  tons. 

16.  If  5  men,  by  laboring  10  hours  a  day,  can  mow  a  field  of 
30  acres  in  10  days,  how  long  will  it  require  8  men  and  7  boys, 
provided  each  boy  can  do  /y  ^.s  nauch  as  a  man,  to  mow  a  field 
containing  54  acres  ?  '  Ans.  7-rV-  davs. 

17.  If  2  men  can  build  12f  rods  of  wall  in  6^  days,  how 
long  will  it  take  18  men  to  build  247^^^  rods  ?      Ans.  14  days. 

18.  If  248  men,  in  5^  days  of  11  hours  each,  dig  a  trench  of 
7  degrees  of  hardness,  and  23 2 J-  feet  long,  3|  feet  wide,  and  2^ 
feet  deep,  in  how  many  days  of  9  hours  each  will  24  men  dig  a 
trench  of  4  degrees  of  hardness,  and  337^  feet  long,  5f  feet  wide, 
and  3^  feet  deep  ?  Ans.  132  days. 


248  PROFIT   AND   LOSS 


PROFIT    AND    LOSS. 

248.  Profit  and  Loss  is  the  process  by  which  merchants  and 
other  traders  estimate  their  gain  or  loss  in  buying  and  seUing 
goods. 

Gains  and  losses  are  usually  reckoned  on  the  prime  or  first 
cost  of  articles. 

249.  To  find  the  rate  per  cent,  of  profit  or  loss  when ' 
the  cost  and  selling  price  are  given. 

Ex.  1.  If  I  buy  flour  at  $  4  per  barrel,  and  sell  it  at  $  5  per 
baiTcl,  what  is  the  gain  per  cent.  ?  Ans.  25  per  cent. 

OPERATION. 

$5_$4  =  $1;  1=  1.00  -T-  4  =  .25,  or  25  per  cent. 

By  subtracting  the  cost  from  the  selling  price,  we  find  the  gain  per 
barrel  to  be  $  1.  Now,  if  the  gain  is  $  1  on  $  4 ,» it  is  \  of  the  cost, 
and  ^  =  .25,  or  25  per  cent. 

OPERATION   BY  PROPORTION. 

,  $5  — $4=$1;  $4:$1::  1.00  :  .25,  that  is,  25  per  cent. 

2.  If  I  buy  flour  at  $  5  per  barrel,  and  sell  it  at  $  4  per  barrel, 
what  is  the  loss  per  cent.  ?  Ans.  20  per  cent. 

OPERATION. 

$5_$4  =  $1;  i=  1.00  H-  5  =  .20,  or  20  per  cent. 

By  subtracting  the  selling  price  from  the  cost,  we  find  the  loss  per 
barrel  to  be  $  1."  Now,  if  the  loss  is  $  1  on  S  5,  it  is  \  of  the  cost,  and 
^  =  .20,  or  20  per  cent.  From  this  analysis,  and  that  of  the  preceding 
example,  it  is  seen  that  the  operation  is  equivalent  to  making  the  gain 
or  loss  the  numerator  of  a  fraction,  and  the  cost  the  denominator,_and 
then  reducing  this  fraction  to  a  decimal ;  or,  in  short,  to  simply  divid- 
ing the  gain  or  loss  by  the  cost. 

OPERATION  .BY  PROPORTION. 

$5  —  $4  =  $1;  $5:  $1::100  per  cent. :  20  per  cent. 

EuLE  1.  —  Divide  the  gain  or  loss  hj  the  cost,  and  the  quotient  icill  he 
the  gain  or  loss  per  cent.     Or, 

Rule  2.  —  As  the  cost  is  to  the  sum  gained  or  lost,  so  is  100 
p(ir  cent,  to  the  per  cent,  gained  or  lost. 

248  What  is  profit  nnd  loss?  What  is  the  first  rule  for  finding  the  profit 
or  loss  in  buying  or  selling  goods  ?     Wliat  is  the  second  rule  1 


PROFIT   AND   LOSS.  249 

Examples  for  Practice. 

3.  Bought  40  yards  of  broadcloth  at  $  5.-10  per  yard,  and  I 
sell  f  of  it  at  $  6  per  yard,  aiid  the  remainder  at  $  7  per  yard ; 
what  do  I  gain  per  cent.  ?  Ans.  15|^  per  cent 

4.  A  merchant  purchased  for  cash  50  barrels  of  flour,  at  $  5 
per  barrel,  and  immediately  sold  the  same  on  8  mouths'  credit, 
at  $  5.98  per  baiTel ;  what  does  he  gain  per  cent  ? 

Ans.  15  per  cent- 

5.  A  grocer  bought  a  hogshead  of  molasses,  containing  100 
gallons,  at  30  cents  per  gallon  ;  but  30  gallons  liaving  l(-akcd 
out,  he  disposed  of  the  remainder  at  40  cents  per  gallon.  Did  he 
gain  or  lose,  and  how  much  per  cent.  ? 

Ans.  Lost  6§  per  cent. 

6.  A  gentleman  m  Rochester,  N.  Y.,  purchased  3000  bushels 
of  wheat,  at  $  1.12^  per  bushel.  He  paid  5  cents  per  bushel  for 
its  transportation  to  New  York  city,  and  then  sold  it  at  $  1.37^ 
per  bushel ;  what  did  he  gain  per  cent.  ? 

Ans.  17jy  per  cent. 

7.  J.  Morse  bought,  in  Lawrence,  a  lot  of  land  7^\  rods 
square,  for  $  5  per  square  rod.  He  sold  the  land  at  5  cents  per 
square  foot ;  what  did  he  gain  per  cent.  ? 

Ans.  172|-  per  cent. 

250.  To  find  the  selling  price,  when  the  cost  and  the 
gain  or  loss  per  cent,  are  given. 

Ex.  1.  If  I  buy  flour  at  $  4  per  barrel,  for  how  much  must  I 
sell  it  per  barrel  to  gain  25  per  cent.  ?  Ans.  $  5. 

OPEEATION. 

$  4  X  -25  =  $  1.00  ;  then  $4-f-$l=$5,  Ans. 

If  I  sell  the  flour  for  25  per  cent,  gain,  I  sell  it  for  .25  more  than  it 
cost.  Therefore,  if  I  add  to  the  cost  .25  of  the  cost,  the  sum  will  be 
the  price  per  barrel  for  which  the  flour  must  be  sold. 

OrERATION   BY  PEOPORTION. 

■    1.00  -f  .25  =  1.25  ;  1.00  :  1.25  :  :  $  4  :  $  5,  Ans. 

2.  If  I  buy  flour  at  $  5  per  baxTel,  for  what  must  I  sell  it  per 
barrel  to  lose  20  per  cent.  ? 

250.  Explain  how  you  find  the  selUng  price  when  the  cost  and  the  gain  or 
loss  per  cent,  are  given. 


250  PROFIT   AND   LOSS. 

OPERATION. 

$ 5  X  .20  =  1.00 ;  $5  —  $1=$4,  Ans. 

If  I  sell  the  flour  for  20  per  cent,  loss,  1  sell  it  for  .20  less  than  it 
cost.  Therefore,  if  I  subtract  from  the  cost  .20  of  the  cost,  the  re- 
mainder will  be  the  price  per  barrel  for  which  the  flour  must  be  sold. 

OPEHATION.  BY  PROPORTIOX. 

1.00  —  .20  =  .80  ;  1.00  :  .80  : :  $  5  :  $  4,  Ans. 

Rule  1.  —  Find  the  percentage  on  the  cost  at  the  given  rate  per  cent., 
and  add  it  to  the  cost,  or  subtract  it  from  the  same,  according  as  the  selU 
ing  price  is  to  he  that  of  profit  or  loss.     Or, 

Rule  2.  —  As  1  is  to  1  with  the  profit  per  cent,  added,  or  loss  per  cent, 
subtracted,  expressed  decimally,  so  is  the  given  price  to  the  price  required. 

Examples  for  Practice. 

3.  Bought  a  hogshead  of  molasses  containing  120  gallons,  for 
30  cents  per  gallon,  but  it  not  proving  so  good  as  Avas  expected, 
I  am  willing  to  lose  10  per  cent,  on  the  cost ;  what  shall  I  re- 
ceive for  it  ?  Ans.  $  32.40. 

4.  A  grocer  bought  a  hogshead  of  sugar,  weighing  net  8cwt. 
8qr.  51b.,  for  $  88 ;  for  what  must  he  sell  it  per  pound  to  gain 
20  per  cent?  Ans.  12  cents  per  pound. 

5.  J.  Simpson  bought  a  farm  for  S  1728  ;  for  what  must  it  be 
sold  to  gain  12  per  cent.,  provided  he  is  to  wait  8  month?;,  Avithout 
interest,  for  his  pay?  Ans.  $  2012.77-1-. 

6.  J.  Fox  purchased  a  barrel  of  vinegar  containing  32  gal- 
lons, for  $  4 ;  but  8  gallons  having  leaked  out,  for  how  much 
must  he  sell  the  remainder  per  gallon  to  gain  10  per  cent,  on  the 
cost?  Ans.  $0.18^  per  gallon. 

7.  Bought  a  horse  for  $  90,  and  gave  my  note  to  be  paid  in  6 
months,  without  interest ;  what  must  be  my  cash  price  to  gain  20 
per  cent,  on  my  bargain  ?  Ans.  $  104.84-|-. 

8.  H.  Tilton  bought  7cwt.  of  coffee  at  $  11.50  per  cwt.,  but 
finding  it  injured,  he  is  willing  to  lose  15  per  cent.  ;  for  how 
much  must  he  sell  the  7cwt.  ?  Ans.  $  G8.42-|-. 

2,51.  To  find  the  COST  when  the  selhng  price  and  the 
gain  or  loss  per  cent,  are  given.  • 

Ex.  1.  If  I  sell  flour  at  $  5  per  barrel,  and  by  so  doing  make 
25  per  cent.,  Avhat  was  the  cost  of  the  llour? 

Ans.  $  4  per  barrel. 

250.  Wli:it  is  the  first  rule  for  findiiis  nt  wlint  price  goods  must  be  sold 
to  yiiin  or  lose  a  yivcn  per  cent.  ]     The  second  rule  I 


PROFIT   AND   LOSS.  251 

OPERATION. 

$5.00-1-  1.25  =  $4,  Ans. 

Since  the  gain  is  25  per  cent,  of  the  cost,  the  selling  price,  S  5,  is 
equal  to  the  cost  increased  by  25  per  cent,  of  the  cost,  or  to  1.25  of  the 
cost;  Hence,  the  cost  must  be  as  many  dollars  as  1.25  is  contained 
times  in  5,  or  $4. 

OPERATION    BY    PROPORTION. 

1.00  -{-  .25  =  1.25 ;  1.25  :  1.00  :  :  f  5  :  $4,  Ans. 

2.  If  I  sell  flour  at  $4  per  barrel,  and  by  so  doing  lose  20  per 
cent.,  Avhat  was  the  cost  of  the  flour  ?  Ans.  $  5  per  barrel. 

OPERATION.    . 

$4.00^.80  =  $5,  Ans. 

Since  the  loss  is  20  per  cent,  of  the  cost,  the  selling  price,  S  4,  is 
equal  to  th'e  cost  decreased  by  20  per  cent,  of  the  cost,  or  .80  of  the 
cost.  Hence,  the  cost  must  be  as  many  dollars  as  .80  is  contained  times 
in  5,  or  $  5. 

OPERATION    BY   PROPORTION. 

1.00  —  .20  =  .80  ;  .80  :  1.00  :  :  $4  :  $5,  Ans. 

Rule  1.  —  Divide  the  selling  price  by  1  increased  hy  the  gain  per 
cent.,  or  by  1  decreased  by  the  loss  per  cent.,  expressed  decimally,  and 
the  quotient  will  be  the  cost.      Or, 

Rule  2.  —  As  \  tvith  the  gain  per  cent,  added,  or  loss  per  cent,  sub- 
tracted, expressed  decimally,  is  to  \,  so  is  the  selling  price  to  the  cost. 

Examples  for  Practice. 

3.  Having  used  my  chaise  1 6  years,  I  am  willing  to  sell  it  for 
$  80  ;  but  by  so  doing  I  lose  62  J-  per  cent. ;  Avhat  was  the  cost  of 
the  chaise  ?  Ans.  $  213.33^. 

4.  If  I  sell  wood  at  $  7.20  per  cord,  and  gain  20  per  cent., 
what  did  the  wopd  cost  me  per  cord  ?  Ans.  $  6  per  cord. 

5.  J.  Adams  sold  40  cases  of  shoes  for  $1600,  and  gained  18 
per  cent. ;  what  was  the  first  cost  of  the  shoes  ? 

Ans.  $1355.93-f. 

J 

251.  "Wliat  is  the  first  rule  for  finding;  the  cost,  wlien  the  selling  price  and 
the  gain  or  loss  per  cent,  are  given  7     The  second  >^ul«  1 


I  secona  >^ui 


252  PROFIT   AND   LOSS. 

6.  Sold  17  baiTels  of  flour  at  $  8  per  barrel,  for  which  I  re- 
ceived a  note  payable  in  3  months.  This  note  I  had  discounted 
at  the  Granite  Bank,  but,  on  examining  my  account,  I  find  I  have 
lost  10  per  cent,  on  the  flour;  what  was  its  cost? 

Ans.  $148.76+. 

252.  The  selling  price  of  goods  and  the  rate  per  cent, 
being  given,  to  find  what  the  gain  or  loss  per  cent,  would 
be,  if  sold  at  another  jirice. 

Ex.  1.  If  I  sell  flour  at  $  5  per  barrel,  and  gain  25  per  cent., 
what  should  I  gain  if  I  were  to  sell  it  for  $  7  per  barrel  ? 

OPERATION. 

We  find  the  cost  of  the  flour  per  barrel,  as  in  Art.  251.     Thus, 
$  5.00  -^  1.25  ==  $  4.00,  the  cost  per  barreL 

We  find  the  gain  per  cent,  on  the  cost  when  sold  at  S  7  per  barrel, 
as  in  Art.  249.     Thus, 

$7  —  $4  =  $3;  3.00  -^-  4  =  .75,  or  75  per  cent 

OPERATION   BY   PROPORTION. 

1.00-1- .25  =  1.25;  $5  :$7:  :  1.25  :  1.75  ; 
1.75  —  1.00  =  .75,  that  is,  75  per  cent. 

Rule  1.  —  Find  the  cost  (Art.  251),  and  then  the  gain  or  loss  per 
cent,  on  this  cost  at  the  proposed  selling price_.     (Art.  249.)     Or, 

Rule  2.  —  ^,<?  the  Jirst  price  is  to  the  proposed  price,  so  is  1  niih  the 
gain  per  cent,  of  the  Jirst  price  added,  or  the  loss  per  cent,  of  the  Jirst  price 
subtracted,  to  1  with  the  gain  per  cent,  of  the  proposed  price  added,  or 
with  the  loss  per  cent,  of  the  proposed  price  subtracted. 

Note.  —  If  the  result  by  the  last  rale  exeeeds  1.00,  the  excess  is  the  gain 
per  cent. ;  but,  if  it  is  less  than  1.00,  the  delicieucy  is  the  loss  per  cent. 

Examples  for  Practice. 

2.  Sold  a^  quantity  of  oats  at  28  cents  per  bushel,  and  gained 
12  per  cent. ;  what  per  cent,  should  I  gain  or  lose,  if  I  were  to 
sell  them  at  24  cents  per  bushel  ?  Ans.  Lose  4  per  cent. 

3.  S.  Rice  sold  a  horse  for  $  37.50,  and  lost  25  per  cent. ; 
what  would  have  been  his  gain  per  cent,  if  he  l)§d  sold  him  for 
$  75  ?  Ans.  50  per  cent. 

4.  S.  Plielps  sold  a  quantity  of  wheat  for  $  1728,  and    took 

2.')2.  Wliat  is  the  first  vnlc  for  findinp:  what  gain  or  loss  is  marie  by  selling 
goods  at  another  prire  when  the  selling;-  price  and  rate  ]x'r  ceiU.  are  given? 
The  second  rule?  If  ilu-  answer  exceeds  1  00  what  is  the  excess?  If  it  is 
less  than  1.00,  what  is  tiie  delicieucy  \ 


MISCELLANEOUS   EXERCISES.  253 

a  note  payable  in  9  months  without  interest,  and  made  10  per 
cent,  on  his  purchase  ;  what  would  have  been  his  gain  per  cent. 
if  he  had  sold  it  to  James  Wilson  for  $  2000  cash  ? 

Ans.  33-[-  per  cent. 

^  MISCELLANEOUS  EXERCISES. 

1.  A  horse  that  cost  $  84,  having  been  injured,  was  sold  for 
$  75.60  ;  what  was  the  loss  per  cent.  ?  Ans.  10  per  cent. 

2.  Sold  a  horse  for  $  75.60,  and  lost  10  per  cent,  on  the  cost ; 
but,  if  I  had  sold  him  for  $  97.44,  what  per  cent,  should  I  have 
gained  on  the  cost  of  the  horse?  Ans.  16  per  cent. 

3.  M.  Star  sold  a  horse  for  $  97.44,  and  gained  16  per  cent.; 
what  would  have  been  his  loss  per  cent,  if  he  had  sold  the  horse 
foi'  $  75.60,  and  what  his  actual  loss  ? 

Ans.  Loss  10  per  cent.     $  8.40  loss. 

4.  If  I  buy  cloth  at  $  5  per  yard,  on  9  months'  credit,  for  what 
must  I  sell  it  per  yard  for  cash  to  gain  12  per  cent.  ? 

.  Ans.  $  5.35-|-. 

5.  A.  Pemberton  bought  a  hogshead  of  molasses,  containing 
120  gallons,  for  $  40  ;  but  20  gallons  having  leaked  out,  for  wliat 
must  he  sell  the  remainder  per  gallon  to  gain  10  per  cent,  on  his 

.  purchase  ?  '  Ans.  $  0.44. 

6.  H.  Jones  sells  flour,  which  cost  him  $  5  per  barrel,  for 
$  7.50  per  barrel ;  and  J.  B.  Crosby  sells  coffee  for  14  cents  per 
pound,  which  cost  him  10  cents  per  pound ;  which  makes  the 
greater  per  cent.  ?        Ans.  H.  Jones  makes  10  per  cent.  most. 

7.  J.  Gordon  bought  160  gallons  of  molasses,  but  having  sold 
40  gallons,  at  30  cents  per  gallon,  to  a  man  who  proved  a  bank 
rapt,  and  could  pay  only  30  cents  on  the  dollar,  he  disposed  of 
the  remainder  at  35  cents  per  gallon,  and  gained  10  per  cent,  on 
his  purchase  ;  what  was  the  cost  of  the  molasses  ? 

Ans.  $41.45-}-. 

8.  D.  Bugbee  bought  a  horse  for  $  75.60,  which  was  10  per 
cent,  less  than  his  real  value,  and  sold  him  for  16  per  cent,  more 
than  his  real  value  ;  what  did  he  receive  for  the  horse,  and  what 
per  cent,  did  he  make  on  his  purchase  ? 

Ans.  Received  $  97.44,  and  made  28|  per  cent. 

9.  A  merchant  bought  70  yards  of  broadcloth  that  was  If 
yards  wide,  for  $  4.50  per  yard,  but  the  cloth  having  been  wet, 
it  shrunk  5  per  cent,  in  Icngili,  and  5  in  width;  for  what  must 
the-  cloth  be  sold  per  square  yard  to  gain  12  per  cent.  ? 

22  Ans.  $  3.19-{-. 


254  PARTNERSHIP. 


PARTNERSHIP,    OR    COMPANY    BUSINESS. 

253.  Partnership  is  the  association  of  two  or  more  persons  in 
business,  with  an  agreement  to  sliare  the  profits  and  losses. 

Partliers  are  the  persons  associated  in  business.        * 

Company,  or  Firm,  is  the  name  of  the  business  association. 

Capital,  or  Joint  Stock,  is  the  money  or  property  invested  in  the 
company  or  firm. 

The  Dividend  is  the  profit  or  gain  on  the  shares  of  the  capital. 

254.  To  find  each  partner's  share  of  the  profit  or  loss 
when  each  one's  stock  is  employed  the  same  time. 

Ex.  1.  John  Smith  and  Henry  Gray  enter  into  partnership  for 
three  years ;  Smith  puts  in  S  4000,  and  Gray  $  2U00.  They 
gain  $570.     What  is  each.man's  share  of  the  gain  ? 

Ans.  Smith's  gain,  $  380  ;  Gray's  gain,  $  190. 

OPERATION. 

$  4  0  0  0,  Smith's  stock,  f  ooo  ^  2^  Smith's  part  of  the  stock. 
$  2  0  0  0,  Gray's       "       f  Sgg  =  ^,  Gray's  part  of  the  stock. 

$  6  0  0  0,  Whole  stock. 

Then  f  of  $  5  7  0,  the  whole  gain,  =  $  3  8  0,  is  Smith's  share  of 

the  gain. 
And    ^  of  $  5  7  0,      "  "         =  S  1  9  0,  is  Gray's  share  of 

■  the  gain. 

Proof,  $  5  7  0 

Since  %  4000  -f  S  2000  =  S  0000  is  the.  whole  stock,  Smith's  part  of 
the  stork  is  fg-^f  =  1;  and  Gray's  part,  2po_^  =  i-.  Tlien,  since  each 
jilfen's  gain  must  correspond  to  his  stock,  f  of  $5*0,  or  $380,  is 
SiiiiLli'sbhare  of  the  gain ;  and  J-  of  $  570,  or  $190,  is  Gray's  share  of 
the  gain. 

t  OPEKATION   BY   PKOPORTION. 

$G000:$4OO0::$57O:$38O,  Smith's  gain. 
$G000:$2000::$570:$19  0,  Gray's  gain. 

253.  What  is  partncrsliip  ?  What  .nre  the  persons  associated  called  ? 
■\Vliut  is  the  association  called  ?  What  the  property  invested  t  What  the 
prolit  or  loss  1 


PARTNERSJilP.  255 

Rule  1.  —  Multiply  (he  icliole  gain  or  loss  by  the  fractions  denoting 
each  partner's  part  of  the  whole  stock,  and  the  products  will  he  the  respec- 
tive shares  of  the  gain  or  loss  of  each  partner.     Or, 

Rule  2.  —  As  the  whole  stock  is  to  each  partner's  stock,  so  is  the  whole 
gain  or  loss  to  each  partner's  gain  or  loss. 

Examples  for  Practice. 

2.  Three  merchants,  A,  B,  and  C,  engagecl  in  trade.  A  put 
in  $  6000,  B  put  in  $  9000,  and  C  put  in  $  5000.  They  gain 
$  840.     AVhat  is  each  man's  share  of  the  gain  ? 

Ans.  A's  gain  $  252,  B's  gain  $  378,  C's  gain  $  210. 

3.  A  bankrupt  owes  Peter  Parker  $  8750,  James  Dole  $  3G10, 
and  James  Gage  $  7000.  His  effects,  sokl  at  auction,  amount  to 
$  6875  ;  of  this  sum  $  375  are  to  be  deducted  for  expenses,  &c. 
What  will  each  receive  of  the  dividend  ? 

Ans.     Parker,    $  2937.75-J-§a ;    Dole,    $  1212.03^6^2_ ;    Gage, 

$  2350.20f/-i-- 

4.  A  merchant,  failing  in  trade,  owes  A  $  500,  B  $  386,  C 
%  988,  and  D  $  126.  His  effects  are  sold  for  $  100.  What  will 
each  man  receive  ? 

Ans.  A  receives  $  25.00,  B  $  19.30,  C  49.40,  D  $  6.30. 

5.  A,  B,  and  C,  engaged  in  trade.  A  put  in  $  700,  B  put  in 
$300,  and  C  put  in  100  barrels  of  flour.  They  gained  $90; 
of  which  sum  C  took  $  30  for  his  part ;  what  will  A  and  B  re- 
ceive, and  what  was  C's  flour  valued  per  barrel  ? 

Ans.  A  receives  $  42,  B  $  18,  C's  flour  $  5  per  barrel 

255,  To  find  each  partner's  share  of  the  profit  or  loss, 
when  the  stock  is  employed  unequal  times. 

Ex.  1.  Josiah  Brown  and  George  Dole  trade  in  company. 
Brown  put  in  $  600  for  8  months,  and  Dole  put  in  $  400  for  6 
months.  They  gain  $  60.  What  is  each  man's  share  of  the 
gain  ? 

OPERATION. 

$600  X8  =  $4800forl  month. 
$400x6  =  $2400forl  month. 

$7  200 
iag-J  =  2^  Brown's  share  in  the  partnership. 
^4^^  ==  ^,  Dole's  share  in  the  partnership. 
I  of  $  60  =  $  40,  Brown's  gain  ;  ^  of  $  60  =  $  20,  Dole's  gain. 

254.  The  rule  for  finding  the  shares  of  profit  or  loss  when  the  stock  is 
employed  the  same  time  ? 


256  PARTNERSHIP. 

$  600  for  8  months  is  the  same  as  $  600  X  8  =  $  4800  for  1  month, 
because  S  4800  would  gain  as  much  in  1  month  as  $  600  in  8  months ; 
and  S  400  for  0  months  is  the  same  as  $  400  X  6  =  $  2400  for  1  month.' 
The  (juestion  then  is  the  same  as  if  Brown  had  put  in  $  4800  and  Dole 
S  24UU  for  1  month  each.  The  whole  stock  would  then  be  S  4800 -f- 
$  2400  =  $  7200,  and  Brown's  share  of  the  aain  would  be  4|M  =  J2 
of  $  CO  =  S  40  ;  and  Dole's  share,  ^^  =  i  of  $  60  =  $  20. 

OPERATION   BY  PKOPOKTION. 

$4800  $7.200:$4800::$60:$4  0,  B's  share. 
$2400  $7  200  :  $2400::  $60:  $20,  D's  share. 
$7200 

Rule  1.  —  Multiply  each  parttier's  stock  hy  the  time  it  was  in  trade, 
and  consider  each  product  a  numerator,  to  be  written  over  the  sum  of  the 
products,  as  a  common  denominator.  Then  multiply  the  whole  gain  or 
los.'i  by  each  of  these  fractions,  and  the  product  ivill  be  the  respective  shares 
of  the  gain  or  loss  of  each  j)artner.     Or, 

Rule  2.  —  Multiply  each  partner's  stock  by  the  time  it  was  in  trade; 
then,  as  the  sum  of  these  products  is  to  each  product,  so  is  the  whole  gain 
or  loss  to  each  partner's  gain  or  loss. 

Examples  for  Practice. 

2.  A,  B,  and  C  trade  in  company.  A  put  in  $  700  for  5 
months  ;  B  put  in  $  800  for  6  months  ;  and  C  put  in  $  500  for 
10  months.  They  gain  $399.  What  is  each  man's  share  of  the 
gain?         Ans.  A's  gain  $  105,  B's  gain  $  144,  G's  gain  $  150. 

3.  Johnson,  Hyde,  and  Tyler  enter  into  business,  under  tiie 
firm  of  Johnson,  Hyde,  &  Co.  Johnson  put  in  at  first  $  1000, 
and  at  the  end  of  6  months  $  500.  Hyde  put  in  at  first  $  800, 
aii'l  at  the  end  of  4  months  $  400  ;  but,  at  the  end  of  10  montlis, 
lie  wiLiidrcw  $  500.  Tyler  put  in  at  first  $  1200,  and  at  the  end 
of  7  months  $  300,  and  at  the  end  of  10  months  $  200.  At  the 
end  of  the  year  they  found  their  net  gain  to  be  $  1000.  What  is 
each  man's  share  ? 

Ans.  Johnson's  gain  $  348.02f §f ,  Hyde's  $  273.78 /St,  Tyler's 
$  378.1  95Vt- 

4.  George  Morse  hired  of  William  Hale,  of  Haverhill,  a  horse 
and  chaise  for  a  ride  to  Ncwburyport,  for  $3.00,  with  the  privi- 
lege of  one  person's  having  a  seat  with  him.     Having  rode  4 

255.  What  arc  the  rules  for  findinj^  the  shares  of  profit  or  loss  when  tlio 
8t()'k  is  eiiiploye<i  for  unetiuii!  times?  Why  do  you  multijily  cadi  man's 
Block  l)y  tiic  time  it  w;ts  in  trada  1 


PARTNERSHIP.  257 

miles,  lie  took  in  Jolin  Jones,  and  carried  lilm  to  Newburyport, 
and  brought  him  back  to  the  •  place  from  which  he  took  him. 
What  phare  of  the  expense  should  each  pay,  the  distance  from 
Haverhill  to  Newburyport  being  15  miles? 

Ans.  Morse  pays  $  1.90,  Jones  pays  $  1.10. 

5.  J.  Jones  and  L.  Cotton  enter  into  partnership  for  1  year. 
January  1,  Jones  put  in  $  1000,  but  Cotton  did  not  put  in  any 
until  the  first  of  April.  What  did  he  then  put  in,  to  have  an 
equal  share  with  Jones  at  the  end  of  the  year  ? 

Ans.  1333.33^. 

6.  S,  C,  and  D  engage  in  partnership,  with  a  capital  of  $  4700. 
S's  stock  was  in  trade  8  months,  and  his  share  of  the  profits  was 
$96;  C's  stock  was  in  the  firm  6  months,  and  his  share  of  the 
gain  was  $  90  ;  D's  stock  was  in  the  firm  4  months,  and  his  gain 
was  $  80.      Required  the  amount  of  stock  which  each  had  in 


the  firm. 


S's  stock  $  1200. 

Ans.  -i  C's  stock  $  1500. 

D's  stock  $  2000. 


7.  A,  B,  and  C  engage  in  trade.  A  put  in  $  300  for  7  months, 
B  put  in  $  500  for  8  months,  and  C  put  in  $  200  for  12  months ; 
they  gain  $  85  ;  what  share  of  the  gain  does  each  receive  ? 

«  Ans.  A  $  21,  B  $  40,  and  C  $  24. 

8.  A  and  B  engage  in  trade,  with  $  500.  A  put  in  his  stock 
for  5  months,  and  B  put  in  his  for  4  months.  A  gained  $  10, 
and  B  gained  $  12  ;  what  sum  did  each  put  in? 

Ans.  A  $  200,  B  $  300. 

9.  A  and  B  trade  in  company.  A  put  in  $  3000,  and  at  the 
end  of  6  months  put  in  $  2000  more  ;  B  put  in  $  6000,  and  at 
the  end  of  8  months  took  out  $  3000  ;  they  trade  one  year,  and 
gain  $  1080  ;  what  is  each  man's  share  of  the  gain  ? 

s  Ans.  A's  share  is  $  480,  B's  $  600. 

10.  Four  men  hired  a  pasture  for  $  50.  A  put  in  5  horses 
for  4  weeks  ;  B  put  in  6  horses  for  8  weeks ;  C  put  in  12  oxen 
for  5  weeks,  calling  3  oxen  equal  to  2  horses ;  and  D  put  in  3 
horses  for  14  weeks.     How  much  ought  each  man  to  pay  ? 

Ans.  A  $  6.66J,  B  $  16.00,  C  $  13.331,  and  D  $  14.00. 

11.  A,  B,  and  C  contract  to  build  a  piece  of  railroad  for 
$7500.  A  employs  30  men  50  days;  B  employs  50  men  36 
days ;  and  C  employs  48  men  and  1 0  horses  45  days,  each  horse 
to  be  reckoned  equal  to  one  man,  and  he  is  also  to  have  $  112.50 
for  overseeing  the  work.     How  much  is  each  man  to  receive  ? 

Ans.  A  receives  $  1875  ;  B,  $  2250 ;  C,  S  3375. 
22* 


258 


CURBENCIES. 


CURRENCIES. 

256.  The  Currency  of  a  state  or  country  is  its  money  or  cir- 
culating medium  of  ti-ade. 

In  the  United  States,  the  gold,  silver,  and  copper  coins  of  the 
country,  foreign  coins  whose  value  has  been  fixed  by  law,  and 
bank-notes,  redeemable  in  specie,  pass  as  money. 

The  Legal  Tender,  in  this  country,  in  payment  of  debts,  is  gold 
and  silver. 

The  Intrinsic  Valtie  of  foreign  coins  is  their  mint  value,  or 
that  depending  upon  the  weight  find  purity  of  the  metal  of  which 
they  are  made  ;  their  Commercial  Value  is  the  price  they  will 
bring  in  the  mai'ket,  and  then-  Legal  Value  is  that  fixed  by  law. 

The  value  of  some  foreign  coin?,  as  fixed  by  present  laws  of 
the  United  States,  is  shown  in  the  following 


TABLE  OF  FOREIGN  CURRENCIES. 


Pound  Ster.  of  G.  Britain, 
Pound  Ster.  of  Br.  Prov., 
Nova  Scotia,  N.  Bruns., 
Newfoundland,  and  Can., 
Dollar  of  Mexico,  Peru, 
Chili,  and  Cen.  America, 
Specie  Dollar  of  Sweden 

and  Norway, 
Specie  Dol.  of  Denmark, 
Ilix  Dollar  of  Bremen, 
Ilix  Dollar,  or  Thaler,  of) 
Prussia   and    Northern  > 
States  of  Germany,        ) 
Ruble,  silver,  of  Russia, 
Guilder  of  Netherlands, 
Florin  of  Netherlands, 
Florin  of  South  of  Germany, 


S4.84 
4.00 

1.00 


.OG 

.05 
.78f 

.69 

.75 
.40 
.40 

.40  I 


Ounce  of  Sicily, 
Pagoda  of  India, 
Tael  of  China, 
Milrea  of  Portugal, 
Milrea  of  Azores,  ♦ 

Ducat  of  Naples, 
Rupee  of  British  India, 
Marco  Banco  of  Hamburg, 
Franc  of  France  and  Bel., 
Livre  Tournois  of  France, 
Leghorn  Livre, 
Lira  of  Lombard}',  Yene- } 
tiau  Kingdom,  ) 

Lira  of  Tuscany, 
Lii'a  of  Sardinia. 
Real  Plate  of  Spain, 
Real  Vellou  of  Spain, 


S2.40 
1.84 
1.48 
1.12 

.80 


.44 
.35 

.18|- 
.16 

.l'6 

.16 

.10 
.05 


Tlie  legal  currency  of  this  country,  jirevious  to  1786,  was 
sterling  money,  or  that  of  pounds,  shillings,  and  j/cnce.  On  the 
adoption  of  the  currency  of  dollars  and  cents,  there  were  in 
circulatiim  c-olonial  notes,  or  bills  of  credit,  which  had  depre- 
ciated in  value.     This  depreciation  being  greater  in  some  scc- 


256.  What  !<?  currency  ?  Wliat  pass  for  monoy  in  tlie  United  States  ? 
What  is  the  intrinsie  Value  of  fon-i<rn  eoins  ?  The  eommereial  vahie  ? 
The  Icfral  value'?  Mention  some  of  the  foreiixn  coins  whose  vahie  has  licen 
fixo<l  hy  law.     What  was  tlie  curi-cuey  of  this  country  previous  io  l  ThO  ? 


REDUCTION. 


259 


tions  than  in  others,  gave  rise  to  the  variation,  in  the  States,  as  to 
the  number  of  shillings  equivalent  to  a  dollai',  as  shown  in  the 
foUowinsc 


1  in 


'■  New  Eng.  States,' 
Virginia, 


TABLE. 

=  6s.  =  Jq  £,  called  New  Eng.  currency ; 
of  which  l£  ==  S  3-1- ;   Is.  =  l(if  cts. 


1  ui 


1  in^ 


1  in 


Kentucky, 

Tennessee, 
C  New  York, 
I   Ohio, 
)  IVIichigan, 
1^  North  Carolina, 

Pennsylvania, 

New  Jersey, 

Delaware, 
,  Maryland, 

j  Georgia,  )=  4s.  8d.=-^£,  called  Georgia  currency ; 

'    South  Carolina,    \  of  which  l£  =$4|-;  Is.  =  214  cts. 
r  Canada,  "I 

-   .    J  Nova  Scotia,         I  =5s.  =  -i-£,  called  Canada  currency;  of 
1  New  Brunswick,  [which  l£  =  $  4  ;  Is.  =  20  cts. 
[_  Newfoundland,    J 

(  5  ==iJ^^^s.  =  -^T^\S,,cal\(:d  English  or   Ster- 

1  in -<  Great  Britain,      [-ling  money;    of  which  l£  =  $4.84;  Is. 


8s.  =  f  £,  called  New  York  currency ; 
"of  which  l£  =  S  21 ;   Is.  =  12-i-  cts. 


=  7s.  Cd.  =  I  £,  called  Pennsylvania  cur- 
rency ;   of  which  1  £  =  $  2f ;  Is.  =  1 3^  cts. 


S~ 


24-1-  ets. 


The  old  currencies  of  tliG  States  are  no  longer  used  in  keeping  accounts, 
yet  the  price  of  articles  is  still  named  by  some  traders  in  the  old  currency 
of  their  State. 

REDUCTION. 

257.  Reduction  of  CarrctlCics  is  the  process  of  finding  the  value 
of  the  denominations  of  one  currency  in  the  denominatious  of 
another. 

258.  To  reduce  old  State  currencies  to  United  States 
money. 

Ex.  1.  Reduce  18£  15s.  6d.  New  England  cuiTency  to  United 
States  money.  Ans.  $  G2.58^, 

We  first  reduce  the  shillings  and 
pence  to  the  decimal  of  a  pound 
(Art.  188),  and  annex  it  to 
the  pounds;   we    then   divide  the 


OPERATION. 

18£  15s.  6d.  =  18.775£. 


18.775  -^i 


TT 


$  62.58A. 


2.'i6.  What  gave  rise  to  the  variation  in  the  old  currency  of  this  country! 
Ki-peat  the  table.  How  are  the  old  curreueics  of  the  States  now  used? 
257.  What  is  reduction  of  currencies  ? 


260  •  CURRENCIES. 

sum  by  ^3^,  because  6s.,  or  a  dollar  in  this  currency,  is  Jg  of  a  pound. 
Hence  the 

Rule.  —  Divide  the  given  sum  expressed  in  pounds  and  decimals  of  a 
pound,  by  the  value  of  %1  expressed  in  a  fraction  of  a  pound.  The  quo- 
tient will  be  the  value  in  dollars. 

Examples  for  Practice. 

2.  Change  144£  7s.  Gd.  of  the  old  New  England  currency  to 
United  States  money.  Ans.  $  481.25. 

3.  Change  74£  Is.  Gd.  of  the  old  currency  of  New  York  to 
United  States  money.  Ans.  $  185. 18f. 

4.  Change  129£  of  the  old  currency  of  Pennsylvania  to 
United  States  money.  Ans.  $  344. 

5.  Change  84£  of  the  old  currency  of  South  Carolina  to 
United  States  money.  Ans.  $360. 

6.  Change  144£  4s.  of  Canada  and  Nova  Scotia  currency  to 
United  States  money.  Ans.  $  576.80. 

7.  Change  257£  8s.  Gd.  English  or  sterling  money  to  United 
States  money.  ^  Ans.  $  1245.937. 

259t  To  reduce  United  States  money  to  old  State  cur- 
rencies. 

Ex.  1.  Reduce  $  152.625  to  old  New  England  currency. 

Ans.  45£  15s.  9d. 

OPEKATION.  Since   6s.,  or  a  dollar,  in  this  our- 

152.625  X  T(j  =  45.7875£.     rency,  is  ^^  of  a  pound,  we  nuilti])ly 

45.7875£  =:  45£  15s.  9d.      ^^%  S'''';"  '"!"  .^>'  ^  ^'"^^^'/i?"  I'o'  a"^} 

reduce  the   decimal   to  shillings   and 

pence.     (Art.  189;) 

Rule.  —  Midtiph/  the  given  sum  expressed  in  dollars  hy  the  value  of 
$  1  expressed  in  a  fraction  of  a  pound.  The  product  will  he  the  value  in 
pounds. 

Examples  for  Practice. 

2.  Change  $  481.25  to  the  old  currency  of  New  England. 

Ans.  144£  7s.  6d. 

2.59.  How  do  you  reduce  United  States  nioiioy  to  pounds,  sliillings, 
pence,  and  f;utliin};s.  New  Kiif;l;iiid  currency  7  Why  inuhiply  hy  -i\^£  ? 
How  would  you  reduce  United  States  money  to  pounds,  &c.,  Oliio  cur- 
rency ?  How,  to  Pennsylvania  currency  ?  The  general  rule  ?  The  rule 
for  reducinjj  United  States  money  to  pounds,  shillings,  pence,  and  farthings, 
of  the  dilFcrcnt  currencies  ? 


EXCHANGE.  261 

f 

3.  Change  $  185.18|  to  tlie  old  currency  of  New  York. 

Ans.  74£  Is.  6d. 

4.  Change  $  344  to  the  old  currency  of  Pennsylvania. 

Ans.  129£. 

5.  Change  $360  to  the  old  currency  of  South  Carolina. 

Ans.  84£. 

6.  Change  $  576.50  to  Canada  and  Nova  Scotia  currency. 

Ans.  144£  2s.  6d. 

7.  Change  $  1245.937  to  English  or  sterling  money. 

Ans.  257£  8s.  6d. 

260«  To  reduce  any  foreign  currency  to  United  States  money, 
and  United  States  money  to  any  foreign  currency,  when  the 
value  of  a  unit  of  the  currency  is  known  (Art.  256). 

Multiply  or  divide,  as  the  case  may  require,  by  the  inlue  of  the  unit  of 
the  given  currency  expressed  in  United  States  money. 


Examples  for  Practice. 

Ex.  1.   Eeduce  123  rubles,  silver,  of  Eussia,  to  United  States 
money.  Ans.  $  92.25. 

2.  Reduce  $  27.90  to  francs.  Ans.  150  francs. 

3.  What  is  the  value  of  121   thalers   of  Prussia  in   United 
States  money  ?  Ans.  %  83.49. 

4.  What  is  the  value  of  $  165.20  in  florins  ? 

Ans.  413  florins. 

5.  A  merchant  purchased  tea  in  China  to  the  amount  of  216 
taels.     What  did  it  cost  in  United  States  money  ? 

Ans.  $  319.68. 

6.  How  many  reals,  plate  of  Spain,  are  equal  to  $  5137.90  ? 

Ans.  51379. 


EXCPIANGE 


261  •  Exchange,  in  commerce,  is  the  paying  or  receiving  of 
money  in  one  place  for  an  equivalent  sum  in  another,  by  means 
of  Drafts  or  Bills  of  Exchange. 

260.  How  do  you  reduce  any  foreign  currency  to  United  States  money,  and 
United  States  money  to  any  foreign  cuiTency  1  — 261.  What  is  exchange  ? 


262  EXCHANGE. 

A  Bill  of  Exchauge  is  a  written  order,  to  ?;ome  person  at  a  dis- 
tance, to  pay  a  certain  sum,  at  an  appointed  time,  to  another 
person,  or  to  liis  order. 

The  Malier  or  Drawer  of  a  bill  is  the  person  who  draws  it. 

The  Buyer,  Taker,  or  Remitter  of  a  bill  is  the  person  for  whom 
it  is  drawn. 

The  Drawee  is  the  person  on  whom  it  is  di'awn ;  who  is  also 
called  the  acceptor,  after  he  has  accepted  it. 

The  Indorser  of  a  bill  is  the  person  who  indorses  it. 

The  Holder  or  Possessor  of  a  bill  is  the  person  in  whose  legal 
possession  the  bill  may  be  at  any  time. 

Exchange  is  at  par  when  a  certain  sum,  at  the  place  from 
which  it  is  remitted,  will  pay  an  equal  sum  at  the  place  to  which 
it  is  remitted. 

It  is  said  to  be  at  a  premium,  or  dbot^e  par,  when  the  balance 
of  trade  is  against  the  place  from  which  the  bill  is  remitted  ;  and 
beloiv  par  when  the  balance  of  trade  is  in  favor  of  the  place  from 
which  the  bill  is  remitted. 

INLAND    BILLS. 

262.  An  Inland  Bill  of  Excliangc,  or  Draft,  is  one  of  which  the 

drawer  and  drawee  ai'e  both  residents  of  the  same  country. 

263.  To  find  the  cost  of  an  inland  bill,  or  draft. 

Add  to  the  face  of  the  Ml,  or  draft,  the  amount  of  premium,  or  sub- 
tract from  the  face  of  the  bill,  or  draft,  the  amount  of  discount. 

Examples  for  Practice. 

Ex.  1.  What  is  the  cost  of  the  following  bill  of  exchange,  or 
draft,  at  1^  per  cent,  discount  ?  Ans.  $  445.22. 

$452.  Boston,  March  6,  1856. 

At  sight,  fay  to  William  Dura,  or  order,   four  hundred  and 
ffty-two  dollars,  value  received,  and  charge  the  same  to  my  ac- 
count. Edwin  Danton. 
To  Lewis  Fontenvt, 

Merchant,  New  Orleans. 
2.  A  merchant  in  Chicago  purchased  a  bill  on  New  York  for 
$  1164,  at  1  per  cent,  premium  ;  what  did  he  pay? 

Ans.  $1175.64. 

2f)l.  What  is  a  bill  of  exchange'?  Who  is  the  niaktr,  or  drawer,  of  a 
hiin  Tlie  buyer,  taker,  or  remitter  ?  Tiie  drawee  7  The  indorser  ?  Tho 
bolder,  or  possessor"?  When  is  cxehaii<,rc  at  pari  AVhcii  at  a  premiiiiu  ? 
When  at  a  discount?  —  2(J2.  Whall:  is  an  inland  bill,  or  draft  ?  —  2G3.  How 
do  you  find  the  value  of  an  inland  bill,  or  draft? 


FOREIGN   BILLS.  263 

3.  "What  costs  a  bill  on  Burlington,  Iowa,  for  $  4000,  at  2}  per 
cent,  discount?  Ans.  $3900. 

4.  What  costs'  a  bill  on  Buffalo  for  $  450,  at  f  of  1  per  cent, 
discount?  Ans.  $447.1 8f. 

5.  What  costs  a  draft  from  the  Girard  Bank,  Philadelphia,  on 
the  Bank  of  Commerce,  Boston,  for  $2517.70,  at  |- of  1  per 
cent,  premium  ?  Ans.  $  2520.84-|-. 

FOEEIGN  BILLS. 

26 i.  A  Foreign  Bill  of  Exchange  is  one  of  which  the  drawer 
and  drawee  are  residents  of  dilierent  countries. 

Foreign  bills  are  usually  drawn  in  sets ;  that  is,  at  the  same 
time  there  are  drawn  two  or  more  bills  of  the  same  tenor  and 
date,  each  containing  a  condition  that  it  shall  continue  payable 
only  while  the  others  remain  >unj)aid. 

Note.  — Each  bill  of  a  set  is  reniitted  in  a  different  manner,  in  order  to 
guard  against  loss  or  delay ;  and  when  one  of  the  set  has  been  accepted  and 
paid,  the  others  become  worthless. 

EXCHANGE   ON  ENGLAND. 

265.  The  exchange  value  in  the  United  States  of  the  pound 
sterling  of  Great  Britain,  is  that  of  its  former  legal  value,  $  4| 
=  $  4^0-  ==  $  4.44|.  Tlie  commercial  value  is  generally  about 
9  per  cent,  more  than  this  exchange,  or  nominal  par  value. 

Thus,  nominal  par  value  being  =  $  4.44|-. 

To  which  we  add  9  per  cent,  premium,      =        .40 


The  commercial  par  value  will  be  =  $  4.84|^. 

Therefore,  jvhen  the  nominal  exchange  between  the  United 
States  and  Great  Britain  exceeds  9  per  cent,  premium,  it  is 
above  commercial  par  ;  when  less,  it  is  below  that  par. 

Note. —  The  intrinsic  or  mint  value  of  a  pound  sterling  according  to  the 
pure  metal  in  an  English  sovereign,  is  $4,861,  so  that  sterling  money  is 
actual  par  when  it  is  quoted  at  9f  per  cent,  premium. 

2^6.    To  find  the  cost  of  a  bill  on  England. 

Ex.  1.  What  should  be  paid  for  the  following  bill  at  9;^  per 
cent,  premium?  Ans.  $4866.G6f. 

264.  What  is  a  foreign  bill  of  exchange  ?  How  are  foreign  bills  usually 
drawn?  Why]  —  26.5.  What  is  the  exchange  value  of  the  pound  sterling 
of  Great  Britain,  in  United  States  money  1  How  does  this  ditfcr  from  the 
commercial  par  value  1 


264  EXCHANGE. 

Exchange  for  £  1000.  New  YorTc,  May  16,  1856. 

Tliirly  days  after  sight  of  this  frst  of  exchange  (second  and 
third  of  the  same  tenor  and  date  unpaid),  pay  J.  W.  Hathaioay 
S^  Co.,  or  order,  London,  one  thousand  pounds  sterling,  value 
received,  and  place  the  same  to  my  account. 

To  Bates,  Barixg,  &  Co.,  London.  Rufus  W.  King. 

OPERATION. 

$^  X  1.095  =  $ 4.86G| ;  1000  X  4.8G6|  =  $ 4866.66f. 

We  multiply  $-*^,  the  nominal  value  of  a  pound,  by  1.095,  the 
given  rate,  decimally  expressed,  and  obtain  $  4.866|  as  the  cost 
of  a  pound  at  that  r&te;  and  1000  multiplied  by  the  number 
denoting  this  cost  gives  $  4866.66f  as  the  cost  of  the  bill. 

Rule.  —  Multiply  the  face  oJ*tlie  bill  by  the  cost  of  one  pound  at  the 
given  rate  of  exchange,  and  the  product  will  be  the  cost  in  dollars. 

KoTE.  —  Wlien  there  are  in  the  given  sum,  shillings,  pence,  or  farthings, 
they  must  be  reduced  to  a  decimal  of  a  pound. 

Examples  for  Practice. 

2.  A  merchant  in  Boston  wishes  to  purchase  a  bill  of  572£ 
10s.,  on  Liverpool,  the  premium  being  8J-  per  cent. ;  what  will  it 
cost  him  in  dollars  and  cents  ?  Ans.  $  2760.72f . 

3.  If  J.  C.  Sherman,  of  Chicago,  should  remit  to  London 
1200£,  exchange  being  at  9^  per  cent.,  what  will  be  the  cost  of 
the  bill  in  United  States  money?  Ans.  $  5826.6Gf. 

267.  To  find  the  face  of  a  bill  on  England,  which  can 
be  purchased  for  a  given  -sum. 

Ex.  1.  When  exchange  is  at  9^  per  cent,  premium,  what  will 
be  the  amount  of  a  bill  on  London  which  I  can  purchase  for 
$  4866.66§  ?  Ans.  1000£. 

OPERATION. 

$  Y  X  1.095  =  $4.866§  ;  48G6.66| -4-  4.866§  =  1000£. 


206.  The  rule  for  findinfj  the  cost  of  a  bill  on  Enjrland  in  United  States 
currency  ?  —  267.  The  rule  for  findinjr  the  fiice  of  a  bill  on  England,  which 
can  be  purchased  for  a  given  sum  oT  United  States  currency  1 


EXCHANGE.  2G5 

We  find,  as  by  Art.  266,  the  cost  of  one  pound  at  the  given 
rate  of  exchange.  The  given  sum,  $  48G6.66f,  we  divide  by 
the  cost  of  one  pound,  and  obtain  1000£  as  the  required  face  of 
the  bill. 

Rule.  —  Divide  the  given  sum  by  the  cost  of  one  poun'd  at  the  given 
rate  of  exchange,  and  the  quotient  will  he  the  face  of  the  hill  in  pounds. 

Examples  for  Practice. 

2.  J.  Reed,  of  Cincinnati,  proposes  to  make  a  remittance  to 
Liverpool  of  $  1640,  exchange  being  at  8^  per  cent,  premium; 
what  will  be  the  face  of  the  bill  he  can  remit  for  that  sum  ? 

Ans.  340£  Is.  lOd. 

3.  A  merchant  wishes  to  remit  $  500  to  England,  exchange 
being  at  10  per  cent,  premium ;  what  will  be  the  face  of  the  bill 
he  can  purchase  for  that  sum  ?  Ans.  102£  5s.  5d.-{-. 

EXCHANGE  ON  FRANCE. 

268.  In  France  accounts  are  kept  in  francs  and  centimes. 
The  centimes  are  hundredths  of  a  franc.  All  bills  of  exchange 
on  France  are  drawn  in  francs,  and  are  bought,  sold,  and  quoted 
as  at  a  certain  number  of  francs  to  the  dollar. 

269.  To  find  the  cost  of  a  bill  on  France. 

Divide  the  face  of  the  bill  by  the  cost  of  one  dollar  in  francs,  and  the 
quotient  will  be  the  cost  in  dollars. 

Examples  for  Practicf.. 

Ex.  1.  "Wliat  must  be  paid,  in  United  States  currency,  for  a 
bill  on  Paris  of  2380  francs,  exchange  being  5.15  francs  per  dol- 
lar ?  Ans.  $  462.13+. 

2.  How  many  dollars  will  purchase  a  bill  on  Havre  of  30000 
francs,  exchange  being  5. 17 J-  francs  per  dollar? 

Ans.  $5797.104-. 

3.  AYhat  is  the  cost  of  a  bill  on  Paris  of  62500  francs,  ex- 
change being  5.12  francs  per  dollar?  Ans.  $  12207.03-j-. 

270.  To  find  the  face  of  a  bill  on  France,  which  can  be 
purchased  for  a  given  sum. 

268.  How  are  accounts  kept  in  France  ?  How  are  all  bills  of  exchange 
on  France  drawn  ?  —  260.  How  do  you  find  the  cost  in  United  States  cur- 
rency of  a  hill  on  France?  —  270.  How  do  you  find  the  face  of  a  bill  on 
France,  which  can  be  purchased  for  a  given  sum  of  United  States  money  ? 


266  DUODl^CIMALS. 

Multiply  the  gicen  sum  hj  the  cost  of  one  dollar  in  francs,  and  the 
product  will  be  the  face  of  the  bill  in  francs. 

Ex.  1.  Allred  Walker,  of  New  York,  pays  $  2500  for  a  bill 
on  Paris,  exchange  being  5.12  francs  per  dollar.  AVhat  was  the 
face  of  the  bill  in  francs?  Ans.  12800. 

2.  When  exchange  on  France  is  at  5.13  francs  per  dollar,  a 
bill  of  how  many  francs  should  $  700  purchase  ?       Ans.  3591. 

3.  Morton  and  Blanchard,  of  Boston,  wish  to  remit  $  G75  to 
Paris,  exchange  being  5.16  francs  per  dollar;  what  Avill  be  the 
face  of  the  bill  of  exchange  they  can  purchase  with  the  money  ? 

Ans.  3483  francs. 


DUODECIMALS. 

271.  Duodecimals  are  a  kind  of  compound  numbers  in  which 
the  unit,  or  foot,  is  divided  into  12  equal  parts,  and  each  of  these 
parts  into  12  other  equal  parts,  and  so  on  indefinitely;  thus,  Jj, 

Duodecimals  decrease  from  left  to  right,  according  to  a  scale 
of  12  (Art.  82  ;  nolo).  The  diiferent  orders,  or  denominations 
are  distinguished  from  each  other  by  accents,  called  indices  placed 
at  the  right  of  the  numerators.  Hence  the  denominators  are  not 
expressed.     Thus, 

1  inch  or  prime,  equal  to     -^     of  a  foot,  is  written  1  in.  or  1'. 

1  second  "  Y^^  "  "  1' 

1"  third  "  .  yy'^^  "  "  1 

1  fourth  «  ^"  «  «  1 


\" 


III 


TABLE. 


nil 


12  fourths  make  V". 

12  thirds  "      1". 


12  seconds  make  1'. 

12  inches  or  primes      "      Ift. 


Note. —  The  foot  expresses  12  linear  inclics,  144  square  inches,  or  1728 
cubic  indies,  according  to  the  measure  in  wiiich  the  duodecimal  is  cmploved. 

ADDITION   AND   SUBTRACTION. 

272.    Duodecimals  arc  added  and  subtracted   in  the 
same  manner  as  compound  numbers. 

271.  AVhat  are  duodecimals?  How  do  duodecimals  decrease  from  left  to 
right  t  How  arc  the  diiUrent  denominations  distinguished  from  each  other  1 
■ — 272.  How  are  duodooimals  added  and  suhtraetodi 


MULTIPLICATION   AND   DIVISION.  267 

Examples  for  Practice. 

1.  Add  together  12ft.  6'  9",  14ft.  7'  8",  165ft.  11'  10". 

Ans.  193ft.  2'  3". 

2.  Add  together  182ft.'ll'  2"  4'",  127ft.  7'  8"  11'"  291ft.  5' 
11"  10"'.  Ans.  602ft.  0'  11"  1'". 

3.  From  204ft.  7'  9"  take  114ft.  10'  6".        Ans.  89ft.  9'  3". 

4.  From  397ft.  9'  6"  11'"  7""  take  201ft.  11'  7"  8'"  10"". 

Ans.  195ft.  9'  11"  2'"  .9"". 

MULTIPLICATION  AND   DIVISION. 

273i  The  denomination  of  the  product  of  any  two  duo- 
decimals. 

Ex.  1.   What  is  the  product  of  9ft.  muhiphed  by  3ft.? 

Ans.  27ft. 

OPERATION. 

9ft.  X  3  =  27ft.      ^^^ 

2.  What  is  the  product  of  7ft.  multipMed  b,y^6'?'    Ans.  3ft.  6'. 

OPERATION. 

6'  =  T?  of  a  foot;   then  7ft.  X  i%-  =  tf  ==  42' ;  42'  -r-  12 

==  3ft.  6'. 

3.  What  is  the  product  of  5'  multiplied  by  4'?      Ans,  1'  8". 

OPERATION. 

5'  =  T^j,  and  4'=  ^^ ;  then  ^-^  X  t%  =  Af  =  20" ;  20"=  1'  8". 

4.  What  is  the  product  of  9'  multiplied  by  11'"  ? 

Ans.  8'"  3"". 

OPERATION. 

9'  =  x'j»and  11'"  =^^^^5  then  j%  X  ^|i^  =  ^^/^  =  99"" ; 

99'"'  -M2  =  8'"  3"". 

Thus,  feet  multiplied  by  a  number  denoting  feet  produce  feet ;  feet, 
by  primes  produce  primes ;  primes,  by  primes  produce  seconds,  &c. ; 
and  the  several  products  are  of  the  same  denomination  as  denoted  by 
the  sum  of  the  indices  of  the  numbers  multiplied  together.     Hence, 

Wien  two  numbers  are  multiplied  together,  the  su7n  of  their  indices 
annexed  to  their  product  denotes  its  denomination. 

271.    To  multiply  duodecimals  together. 

Ex.  1.   Multiply  8ft.  6in.  by  3ft.  7in.  Ans.  30ft.  5'  6". 

273.  Hon-  is  the  denomination  of  the  product  denoted  when  duodecimals 
are  muhiplied  together  1 


268  DUODECIMALS. 

OPERATION.  "^6  ^^^^  multiply  each  of  the  terms  in  the  multi- 

Qc         ni  plicand  by  the  7'  in  the  multiplier ;  thus,  6'  X  ''  = 

°^^'       ^^  42"  =  3'  and  G".     Writing  the  6"  below,  one  place 

^ft-       ^  to  the  right,  we  add  the  3'  to  the  product  of  8ft.  X  7' 

4ft    11'  6"  =59' = -1ft.  and  11',  which  we  write  down.    We  then 

2  5ft'       Qi  multiply  by  the  3fk.,  thus:    6'  X  3  =  18'  =  1ft.  and 
■ 6'.      We  write   the   6'  under   primes  in   the  other 

3  Oft.       5'   6"  partial  product,  and  add  the  1ft.  to  the  product  of 

the  Sft.   X   3,  making  25ft,  which   we  write  down. 
The  partial  products  being  added,  we  obtain  30ft.  5'  6". 

Note. — The  notation  of  feet,  primes,  seconds,  &c.,  of  the  multiplier  is 
retained  in  the  operation  to  note  the  ditfercnt  order  of  units. 

Rule.  —  Write  the  multiplier  under  the  multiplicand,  so  that  the  same 
denominations  shall  stand  in  the  same  column. 

Beginning  at  the  right  hand,  multiply  each  term  in  the  multiplicand  by 
each  term  of  the  multiplier,  and  give  each  term  of  the  product  the  proper 
index,  observing  to  carry  1  for  every  12  from  each  lower  denomination 
^o  tha^next  higher. 

The  sum  of  the  sqismhfartial  products  will  be  the  product  required. 

<^       ""*  -*•  "■^j^l^ft.ES    FOR    PkACTICE. 

"•"^  2rMultip'}y  Sft.  Sin. "by  7ft.  9in.      .  —      Ans.  63ft.  11'  3". 
"'»      3.  Multiply  12ft.  9'  by  Oft.  ll'*r>5^      Ans.  126ft.  5'  3". 

4.  My  garden  is  18  rods  long  and  l^i^rod?"  jvide  ;  a  ditch  is  dug 
round  it  2  feet  wide  and  3  feet  deep  ;  bivl^P^,  ditch  not  being  of 
a  sufficient  breadth  and  depth,  I  have  caused  %  to  "be  dug  1  foot 
deeper,  and,  outside,  1ft.  6in.  wider.  How  many^Jj^^j^eet  will  it 
be  necessary  to  remove  ?  ■  — ^  Ans.  7540. 

5.  I  have  a  room  12  feet  long,  11  feet  wide,  and  7|flot  high. 
la  it  are  two  doors,  6  feet  6  inches  high,  and  30  inches  wide, 
and  the  mop-boards  are  8  inches  high.  There  are  3  windows,  3 
feet  6  inches  wide,  and  5  feet  6  inches  high ;  how  jnany  square 
yards  of  paper  will  it  require  to  cover  the  walls  ? 

Ans.  25^2^\  sq.  yd. 

275.    To  divide  one  duodecimal  by  another. 
Ex.  1.   A  certain  aisle  contains   68ft.  10'  8"  of  floor.     The 
width  of  the  floor  being  2ft.  8',  what  is  its  length  ? 

Ans.  25ft.  10'. 
OPERATION.  We  first  divide  the  68ft.  by 

2ft.  8'  )  6  Sft.   1  0'  8"  (  2  5ft.   1  0'      the  divisor,   and   obtain    25ti 
g  gff        g/  for  the  quotient.    "W^e  multiply 

'- the  entire  divisor  bv  the  25, 

•  2ft.      2'  8"  and  subtract  the  product,  6 Gft. 

2ft.       2'  8"  8',  from  the  correspontling  por- 

tion  of  the  dividend,  and  ob- 


274.  The  rule  for  the  multiplication  of  duodecimals  ? 


I]S"\'OLUTION.  269 

tain  2ft.  2',  to  which  remainder  we  bring  down  the  8",  and  dividing, 
we  obtain  10'  for  the  quotient.  Multiplying  the  entire  divisor  by  the 
10',  we  obtain  2ft.  2'  8",  which  subtract  In  like  manner  as  before, 
leaves  no  remainder.     Therefore,  25ft.  10'  is  the  length  of  the  aisle. 

Rule.  —  Find  how  many  times  the  highest  term  of  the  dividend  icill 
contain  the  divisor.  By  this  quotient  midiiply  the  entire  divisor,  and 
subtract  the  product  from  the  corresponding 'terms  of  the  dividend.  To 
the  remainder  annex  the  next  denomination  of  the  dividend,  and  divide  in 
like  manner  as  before,  and  so  continue  till  the  division  is  complete. 

Examples  fob  Practice. 

2.  What  must  be  the  length  of  a  board,  that  is  1ft.  9in.  wide, 
to  contain  22ft.  2in.  ?  .       Ans.  12ft.  8in. 

3.  I  have  engaged  E.  Holmes  to  cut  me  a  quantity  of  wood. 
Il  is  to  be  cut  4ft.  Gin.  in  length,  and  to  be  "  corded  "  in  a  range 
256ft.  long.  Required  the  hight  of  the  range  to  contain  75 
cords.  Ans.  8ft.  4in. 


INVOLUTION. 

276i  InTOlution  is  the  process  of 'finding  any  power  of  a  num- 
ber. ■ 

A  Power  of  a  number  is  the  product  obtained  by  taking  the 
number,  a  certain  number  of  times,  as  a  factor.  The  factor,  thus 
taken,  is  called  the  root,  or  the  Jirst  power. 

The  Index  or  Exponent  of  a  power  is  a  small  figure  placed  at 
the  right  of  the  root,  indicating  the  number  of  times  it  is  taken 
as  a  factor.  Thus,  6^  indicates  the  second  power  of  6 ;  4?,  the 
third  power  of  4  ;  and  (§)'',  the  fourth  power  of  §. 

The  second  power  of  a  number  is  sometimes  called  its  square  ; 
the  third  power,  its  cube  ;  and  the  fourth  power,  its  bi-quadrate. 

277.   To  raise  a  number  to  any  required  power. 

3  ==      3,  the  first  power  of  3,  written  3  or  3' 

3x3=      9,  the  second  power  of  3,  written      3^ 

3X3X3=   27,  .th^^third  power  of  3,         "  3' 

3X3X3X3=    81,  theTourth  power  of  3,       «  3* 

3X3X3X3  X.3=  243,  the  fifth  ^jower  of  3,  « 

275.  Th%.«»le?— 276.  What  \s  Involution?     A   power?     What  is    the 
number  called  that  denotes  the  power  ?     Where  is  it  placed  1  —  277.  To 
whq,t  is  the  index  in  each  power  equal  ? 
23* 


270  INVOLUTION. 

By  examining  the  several  powers  of  3  in  the  examples  given,  we 
Bee  that  the  index  of  each  power  is  equal  to  the  number  of  times  3  is 
used  as  a  factor  in  the  multiplications  producing  the  power,  and  that 
the  number  of  times  the  number  is  multiplied  into  itself  is  one  less 
than  tlie  power  denoted  by  the  index.     Hence  the 

Rule.  —  Multiply  the  given  number  hy  itself,  as  many  times  less  1,  as 
tJiere  are  units  in  the  exponent  of  ike  required  power. 

Note  1.  —  A  fraction  may  be  raised  to  any  power,  by  involving  its  terms. 
Thus,  the  second  power  of  f  is  f  X  f  =  jV 

Note  2.  —  A  mixed  number  may  be  either  reduced  to  an  improper 
fraction,  or  the  fractional  part  reduced  to  a  decimal,  and  then  raised  to  the 
required  power. 

Examples  for  Practice. 

1.  What  is  the  2d  power  of  6  ?  Ans.  36. 

2.  Wiiat  is  the  third  power  of  5  ?  Ans.  125. 

3.  What  is  the  6th  power  of  4?  Ans.  4096. 

4.  What  is  the  4th  power  of  -J  ?  Ans.  ^\. 

5.  Wliat  is  the  5th  power  of  3f  ?  Ans.  662if  f . 

6.  What  is  the  3d  power  of  .25  ?  Ans.  .015625. 

7.  What  is  the  1st  power  of  17?  Ans.  17. 

r'    278.  To  raise  a  number  to  any  required  power  without 
producing  all  the  intermediate  powers. 

Ex.  1.   What  is  the  8th  power  of  4  ?  Ans.  65536. 

OPERATION. 

12  3  3        "+-        3       -I-         2         —  8 

4,     16,     6  4;     6  4X64X16  =  6553  6. 

We  raise  the  4  to  the  2d  and  to  the  3d  power,  and  write  above  each 
power  its  exponent.  We  then  add  the  exponent  3  to  itself,  and,  in- 
creasing the  sum  by  the  exponent  2,  obtain  8,  a  number  equal  to  the 
power  recpiired.  AVe  next  multiply  G4,  the  power  belonging  lo  the  ex- 
ponent 3,  into  itself,  and  this  product  by  16,  the  power  belonging  to  the 
exponent  2,  and  obtain  G5536  for  the  8th  power. 

Rule.  —  liaise  the  giren  nuinher  to  any  convenient  nuniber  of  powers, 
and  write  above  each  of  the  respective  powers  its  exponent.     Then  add 

277.  The  rule  for  raising  a  nuuili(>r  to  any  required  power?  How  miiy  a 
common  fraction  bo  raised  to  a  rciiiiind  power?  How  a  mixed  uunilitT  ? 
—  278.  Wliat  arc  the  immbers  j)laced  over  the  several  powws  of  4  called, 
and  what  do  they  denote  ? 


EVOLUTION.  271 

■together  such  exponents  as  will  make  a  number  equal  to  the  required  potcer, 
repeating  any  one  when  it  is  more  convenient^  and  the  product  of  the  pow- 
ers belonging  to  these  exponents  will  be  the  required  answer. 


Examples  for  Practice. 

2.  Wliat  is  the  7th  power  of  5  ?  Ans.  78125. 

3.  What  is  the  9th  power  of  6  ?  Ans.  10077696. 

4.  What  is  the  12  power  of  7  ?  Ans.  13841287201. 

5.  Wliat  is  the  8th  power  of  8  ?  Ans.  16777216. 

6.  What  is  the  20th  power  of  4?  Ans.  1099511627776. 

7.  What  is  the  30th  power  of  3  ?  Ans.  205891132094649. 


EVOLUTION. 

279.    Evolntloil  is  the  process  of  finding  the  root  of  a  given 
power.     It  is  the  reverse  of  Involution. 

A  Root  of  a  power  is  a  number  which,  being  muUipKed  into 
itself  a  certain  number  of  times,  will  produce  the  given  power. 
Thus  4  is  the  second  or  square  root  of  16,  because  4X4=16; 
and  3  is  the  third  or  cube  root  of  27,  because  3X3X3=  27. 

Roots  take  the  name  of  the  corresponding  power,  thus  ; 

The  Second,  or  Square  Root,  is  that  of  a  second  power. 

The  Third,  or  Cnbe  Root,  is  that  of  a  third  power. 

The  FoiU'tli,  or  Biqnadrale  Root,  is  that  of  a  fourth  power. 

Rational  Roots  are  those  roots  which  can  be  exactly  found. 

Surd  Roots  are  those  which  cannot  be  exactly  found,  but  ap- 
proximate towards  true  roots. 

Numbers  that  have  exact  roots  are  called  perfect  powers,  and 
all  other  numbers  are  called  imperfect  powers. 

1 JSt . —    . 

278.  The  rule  for  involYing  a  number  without  producing  all  the  inter- 
mediate powers'?  —  279.  What  is  Evohttion  i  Wliat  is  a  root?  From  what 
does  tlie  root  take  its  name  1     Wiiat  are  rational  jools  'i     Surd  root^  I 


:«».: 


272  EVOLUTION. 

Roots  are  denoted  by  writing  the  character  v^,  called  the 
radical  sign,  before  the  power,  with  the  index  of  the  root  over  it, 
or  by  a  tractional  index  or  exponent ;  in  case,  however,  of  the 
second  or  square  root,  the  index  2  is  omitted.  The  third  or  cube 
root  of  27  is  expressed  thus,  /^  27,  or  27?  ;  the  second  or  square 
root  of  25  is  expressed  thus,  \/  25,  or  25*  ;  and  the  square  of  tJie 
cube  root  of  27,  or  the  cube  root  of  the  square  of  27,  is  expressed 
thus,  27i. 

EXTRACTION  OF  THE  SQUAEE  ROOT. 

280.  The  Square  Root,  or  the  root  of  a  second  power,  is 
so  called  because  the  square  or  second  power  of  any  number 
represents  the  contents  of  a  square  surface,  of  which  the  root  is 
the  length  of  one  side, 

281 1  To  extract  the  square  root  of  a  number  is  to  find  such  a 
factor  as,  when  multiplied  by  itself,  will  produce  tlie  given  num- 
ber, or  it  is  to  resolve  the  number  into  two  equal  factors. 

Roots  of  the  first  ten  integers  and  their  squai-es  are :  — 

,    1,       2,       3,       4,       5,       6,       7,       8,       9,       10, 
'  1,       4,       9,     16,     25,     36,     49,     64,     81,     100. 

It  will  be  observed  that  the  second  power  or  square  of  each  of  the ' 
numbers  contains  twice  as  many  figures  as  the  root,  or  twice  as  many 
wanting  one.     Hence,  to  ascertain  the  number  of  figm*es*in  the  square 
root  of  a  given  number, 

Beginning  at  the  right,  point  it  off  into  as  many  2'>eriods  as  possible,  of 
two  fgures  each;  and  there  toUl  be  as  many  fgures  in  the  root  as  there 
are  periods. 

Note.  —  When  the  given  number  contains  an  odd  number  of  figures,  the 
period  at  the  left  can  contaiu  but  one  figure. 

Ex,  1.  I  wish  to  arrange  625  tiles,  each  of  Avhich  is  1  foot 
square,  into  a  square  pavement ;  what  will  be  the  length  of  one 
of  tlie  sides  ?  Ans.  25  feet. 

OPERATION.  We  must  extract  the  square  root  of  G25 

625^25   Ans  to- obtain  one  side  of  the  pavement,  in 

4  feet.     (Art.  280.) 

Beginning  at  the  right  hand,  we  point 

4  5)225  off  the  number  into  periods,  by  placing  a 

2  2  5  point  over  tlie  right-hand   figure   of  each 

period  ;  and  then  find  the  greatest  square 

279.  How  are  roots  denoted  ?  What  is  said  of  the  index  2  ?  —  280.  Wliat 
is  meant  by  the  square  root,  and  why  is  it  so  called?  —  281.  What  is  meant 
by  extracting  the  square  root?  Mow  do  you  ascertain  the  number  of 
figures  in  the  square  root  of  any  number  ? 


EXTRACTION  OF  THE  SQUARE  ROOT. 


273 


number  in  tlie  left-hand  period,  6  (hundreds)  to  be  4  (hundreds),  and 
that  its  root  is  2,  which  we  write  in  the  quotient.  As  this  2  is  in  the 
place  of  tens,  its  value  is  20,  and  represents  the  side  of  a  square, 
the  area  or  superficial  contents  of  which  are  400  square  feet,  as  seen  in 

We  now  subtract  400  feet  from  625  feet, 
and  have  225  feet  remaining,  which  must  be 
added  on  two  sides  of  Fig.  1,  in  order  that  it 
may  remain  a  square.  We  therefore  double 
the  root  2  (tens)  or  20,  one  side  of  the  square, 
to  obtain  the  length  of  the  two  sidus  to  be  en- 
landed,  makinii:  40  feet ;  and  then  inquire  how 
many  tunes  40,  as  a  divisor,  is  contauieu  in 
the  dividend  225,  and  find  it  to  be  5  times. 
This  5  we  write  in  the  quotient  or  root,  and 
also  on  the  right  of  the  divisor,  and  it  repre- 
sents the  width  of  the  additions  E  and  F  to  the 


Fig.  1. 

Fig.  1. 

20  feet. 

D 

(O 

(U 

20 

o 

'Hi 

20 

<? 

o 

trt- 

400 

20  feet. 


square,  as  seen  in  Fig.  2. 


Fig.  2. 
25  feet. 


The  width  of  the  additions  multiplied  by  40,  the  length  of  the  two 
additions,  makes  the  contents  of  the  two  additions  E  and  F  to  be 

200  square  feet,  or  100  feet  for  each.  "  The 
space  G  now  remains  to  be  filled,  to  complete 
the  square,  each  side  of  which  is  5  feet,  or 
equal  to  the  width  of  E  and  F.  We  square 
5,  and  have  the  contents  of  the  last  addition, 
G,  equal  to  25  square  feet.  It  is  on  account 
of  this  last  addition  that  the  la-st  figure  of 
the  root  is  placed  in  the  divisor;  for  we  thus 
obtain  45  feet  for  the  length' of  all  the  addi- 
tions made,  which,  being  multiplied  by  the 
width  (5ft.),  the  last  figure  in  the  root,  the 
product,  225  square  feet,  will  be  the  con- 
tents of  the  three  additions,  E,  F,  and  G, 
and  equal  to  the  feet  rema,ining  after  we 
had  found  the  first  square.  Hence,  we  obtain  25  feet  for  the  length 
of  one  side  of  the  pavement,  since  25  X  25  ==  625,  the  number  of  tiles 
to  be  arranged,  and  equal  to  the  sum  of  the  several  parts  of  Fig.  2 ; 
thus,  400  -}-  100  +  100  -f  25  =  625. 

This  illustration  and  explanation  is  founded  upon  the  principle,  That 


G    I 

5-5 

o 

D 

F 

4i! 

20 

20 

in 

20 

5 

400 

100 

25  feet. 


281.  What  is  first  done  after  dividing  the  number  into  periods?  What 
part  of  Fig.  1  does  this  greatest  square  number  represent  ?  What  place  does 
the  figure  of  the  root  occupy,  and  what  part  of  the  figure  does  it  represent  1 
Why  do  you  double  the  root  for  a  divisor  ?  What  part  of  Fig.  2  does  the 
divisor  represent  f  What  part  docs  the  last  figure  of  the  root  reprcso'^t  ? 
Why  do  you  multiply  the  divisor  by  fhc  last  figure  of  the  root  1  Whnt 
parts  of  the  figure  doc  the  product  represent  ?  W!iy  do  yon  square  the  hist 
fifrure  of  the  root?  Wliat  part  of  the  fijrure  does  this  square  rcnrosont ? 
What  other  way  of  findinp;  the  contents  of  the  additions  without  multiplying 
the  parts  separately  by  the  width  ? 


274  EVOLUTION. 

the  square  of  (lie  sum  of  two  numbers  h  equal  to  the  squares  of  the  num- 
bers, plus  twice  their  product.  Thus,  25  being  equal  to  20  -\-  5,  its 
square  is  equal  to  the  squares  of  20  and  of  5,  plus  twice  the  product  of 
20  and  5,  or  to  400  4-2X20X^  +  25  =  625. 

Rule.  —  Point  off  the  given  number  into  as  many  periods  as  possible 
of  two  figures  each,  by  putting  a  j^oint  over  the  place  of  units,  another 
over  the  place  of  hundreds,  and  so  on. 

Find  the  greatest  square  number  in  the  left-hand  period,  writing  the 
root  of  it  at  the  right-hand  of  the  given  number,  after  the  manner  of  a 
quotient  in  division,  for  the  frst  fgure  of  the  root.  Subtract  this  square 
number  from  the  first  period,  and  to  the  remainder  bring  down  the  next 
period  for  a  dividend. 

Double  the  root  already  found  for  a  divisor,  and  find  how  often  the 
divisor  is  contained  i7i  the  dividend,  exclusive  of  the  right-hand  figure, 
and  annex  the  result  to  the  root  for  the  second  figure  of  it,  and  likewise  to 
the  divisor.*  Multiply  the  divisor  with  the  figure  last  annexed  by  the 
figure  annexed  to  the  root,  and  subtract  the  product  from  the  dividend. 
To  the  remainder  bring  down  the  next  period  for  a  neio  dividend. 

Double  the  root  already  found  fur  a  new  divisor,  and  continue  the  opeV' 
ation  as  before,  till  all  the  periods  have  been  brought  down. 

NoTK  1.  —  The  left-hand  period  may  contain  but  one  figure.  (Art.  281, 
Note.) 

2.  If  the  dividend  does  not  contain  the  divisor,  a  cipher  must  be  phiced  in 
the  root,  and  also  at  the  right  of  the  divisor;  then,  after  bringing  down  the 
next  period,  this  last  divisor  must  be  used  as  the  divisor  of  the  new  divi- 
dend. 

3.  When  there  is  a  rcm.ainder  after  extracting  the  root  of  a  number, 
periods  of  ciphers  may  be  annexed,  and  the  figures  of  the  root  thus  obtained 
•will  be  decimals. 

4.  Tf  tlie  given  number  is  a  decimal,  or  a  whole  number  and  a  decimal, 
the  root  is  extracted  in  the  same  manner  as  in  whole  numbers,  except,  in 
pointing  off  the  decimals,  either  alone  or  in  connection  with  the  whole  num- 
ber, we  place  a  point  over  evciy  second  figure  toward  the  right,  from  the 
separatrix,  and  fill  the  last  period,  if  incomplete,  with  a  cipher. 

.5.  The  square  root  of  any  number  ending  with  2,  3,  7,  or  8,  cannot  be 
exactly  found. 

Examples  for  Practice. 


2.  What  is  the  f=quare  root  of  148996  ? 


*  The  figure  of  the  root  must  generally  be  diminished  by  one  or  two  units, 
on  account  of  the  deficiency  in  enlarging  the  square. 


~m- 


2R1.  Tbo  rule  for  extracting  the  square  root?  What  is  to  be  done  if  the 
dividi'nd  docs  not  contain  the  divisor?  What  must  be  done  if  tlicre  is  a 
remainder  after  extracting  the  root?  What  do  you  do  if  the  given  number 
is  a  decimal  ?     Of  what  numbers  can  the  exact  squ-aro  root  not  be  found  ? 


EXTRACTION   OF   THE   SQUAI?E  ROOT.  275 


OPERATION. 

14899  6(386 
9 


68)589 
544 

766)4596 
4596 

3.  What  is  the  square  root  of  516961  ?  Ana.  719. 

4.  What  is  the  square  root  of  182329  ?  Ans.  427. 

5.  What  is  the  square  root  of  23804641  ?  Ans.  4879. 

6.  Wliat  is  the  square  root  of  10673289  ?  Ans.  3267. 

7.  What  is  the  square  root  of  20894041  ?  Ans.  4571. 

8.  What  is  the  square  root  of  42025  ?  Ans.  205. 

9.  What  is  tlie  square  root  of  1014049  ?  Ans.  1007. 

10.  What  is  the  square  root  of  538?  Ans.  23.194+. 

11.  Wliat  is  the  square  root  of  71  ?  Ans.  8.42 6-j-. 

12.  What  i3  the  square  root  of  7  ?  Ans.  2.645-f-. 

13.  What  is  the  square  root  of  .1024?  Ans.  .32. 

14.  What  is  the  square  root  of  .3364?  Ans.  .58. 

15.  What  is  the  square  root  of  .895  ?  Ans.  .946-|-. 
^16.  Wliat  is  the  square  root  of  .120409  ?  Ans.  .347. 

17.  What  is  the  square  root  of  61723020.96  ?  Ans.  7856.4. 

18.  AVhat  is  the  square  root  of  9754.60423716? 

Ans.  98.7654. 

2<S2i  If  it  is  required  to  extract  the  square  root  of  a  common 
fraction,  or  of  a  mixed  number,  the  mixed  number  must  be  re- 
duced to  an  improper  fraction ;  and  in  both  cases  the  fractions 
must  be  reduced  to  their  lowest  terms,  and  the  root  of  the  nu- 
merator and  denominator  extracted. 

Note.  —  When  the  exact  root  of  tlie  terms  of  a  fraction  cannot  be  found, 
it  must  be  reduced  to  a  decimal,  and  the  root  of  the  decimal  extracted. 

Examples  for  Practice. 

1.  What  is  the  square  root  of  -^^^  ? 

2.  Wliat  is  the  square  root  of  ^||-  ? 

3.  What  is  the  square  root  of  ff f g-  ? 

4.  What  is  the  square  root  of  y^^g-  ? 

5.  What  is  the  square  root  of  "S^y'^  ? 

6.  Wliat  is  the  square  root  of  28f  J  ?  Ans,  5|. 

282.  What  do  you  do  when  it  is  required  to  extract  the  square  root  of  a 
common  fraction,  or  of  a  mixed  number? 


Ans. 

/t- 

Ans. 

il- 

Ans. 

if 

Ans. 

fVt- 

Ans. 

7-|. 

276  EVOLUTION. 

7.  What  is  tlie  square  root  of  47^  J  ?  Ans.  6|. 

8.  What  is  the  square  root  of  ^f  ?  Ans.  .858-|-. 

9.  What  is  the  square  root  of  83f  ?  Ans.  9.14+. 

10.  What  is  the  square  root  of  121J-|  ?  Ans.  11.042+. 

gggS 

11.  What  is  the  square  root  of  — -I  ?  Ans.  f. 

12.  What  is  the  square  root  of l±  ?  Ans.  |. 

1557A 

APPLICATION  OF   THE   SQUARE  ROOT. 

283.  The  square  root  may  be  applied  to  finding  the  dimensions 
and  areas  of  squares,  triangles,  circles,  and  other  surfaces. 

1.  A  general  has  an  array  of  22G576  men  ;  how  many  must  he 
place  rank  and  file  to  form  them  iuto  a  square  ?  Ans.  476. 

2.  A  gentleman  purchased  a  lot  of  land  in  the  form  of  a 
square,  containing  G40  acres ;  how  many  rods  square  is  his  lot  ? 

Ans.  320  rods. 

3.  I  have  three  pieces  of  land;  the  first  is  125  rods  long,  and 
53  wide  ;  the  second  is  G2i  rods  long,  and  34  wide  ;  and  the 
third  contains  37  acres  ;  what  will  be  the  length  of  the  side  of 
a  square  field  whose  area  will  be  equal  to  the  three  pieces  ? 

Ans.  121.11-1- rods. 

4.  W.  Scott  has  2  house-lots  ;  the  first  is  242  feet  square,  and 
the  second  contains  9  times  the  area  of  the  first;  how  many  feet 
square  is  the  second  ?  Ans.  726  feet. 

5.  There  are  two  pastures,  one  of  which"  contains  124  acres, 
and  the  area  of  the  other  is  to  the  former  as  5  to  4 ;  liow  many 
rods  square  is  the  latter?  Ans.  157.48-|- rods. 

6.  I  wish  to  set  out  an  orchard  containing  216  fruit-trees,  so 
that  the  number  of  trees  in  huigth  shall  be  to  the  number  of  trees 
in  l>readtli  as  3  to,  2,  and  the  distance  of  the  trees  from  each  otli- 
<*r  25  feet;  how  many  trees  will  there  be  in  a  row  each  way, 
and  how  many  square  feet  of  ground  will  the  oi'chard  cover  ? 

Ans.  18  in  length  ;  12  in  breadth  ;  116875sq.  ft. 

284.  A  Triangle  is  a  figure  having  three  sides  and  three 
angles. 

A  Riglll-angU'd  Triangle  is  a  figure  having  three  sides  and  three 
angles,  one  of  which  is  a  right  angle. 

283.  To  wlmt  may  the  squaro  root  be  applied?  —  284.  What  is  a  triangle? 
What  is  a  riglit-unglcd  tiiuuglc'? 


APPLICATION  OF  THE  SQUARE  ROOT. 


277 


In  the  triangle,  ABC,  the  angle  at  B  is  a 
right  angle ;  tlie  longest  side,  A  C,  the  hypolh- 
enuse ;  the  side,  A  B,  on  Avhich  the  triangle 
stands,  the  base;  and  the  side,  B  C,  perpen- 
dicular to  the  base,  the  perpendicular. 


Base 


285.  In  every  right-angled  triangle,  the  square  of  the  hypothe- 
nuse  is  equal  to  the  sum  of  the  squares  oj"  the  base  ayid  perpendic- 
ular. 


It  will  be  seen,  by  examining  this  dia- 
gram, that  the  large  square,  formed  on  the 
hypothenuse  A  C,  contains  the  same  number 
of  small  squares  as  the  other  two  counted 


-vsr 

Kx 

y 

1 

B 

together. 


286.  To  find  the  Hypothenuse,  the  base  and  perpen- 
dicular behig  given. 

Rule. —  Add  the  square  of  the  hase  to  the  square  of  the  perpendicu- 
lar, and  extract  the  square  root  of  the  suin. 

287.  To  find  the  Perpendicular,  the  base  and  hypoth- 
enuse being  given. 

Rule. —  Subtract  the  square  of  the  hase  from  the  square  of  the  hy- 
pothenuse, and  extract  the  square  root  of  the  remainder. 

288.  To  find  the  Base,  the  hypothenuse  and  perpen- 
dicular being  given. 

Rule.  —  Subtract  the  square  of  the  perpendicular  from  the  square  of 
the  hypothenuse,  and  extract  the  square  root  of  the  remainder. 

Examples  for  Practice. 

1.  "What  must  be  the  length  of  a  ladder  to  reach  to  the  top  of 
a  house  40  feet  in  bight,  the  bottom  of  the  ladder  being  placed  9 
feet  from  the  sill  ?  An.s.  41  feet. 

284.  What  is  the  longest  side  called  1  W!:at  the  other  two  ?  —  285.  How 
does  the  square  of  the  hypothenuse  compare  with  the  hase  and  perpendicular? 
How  does  this  fact  appear  from  Fig.  2  ■?  —  286.  The  rule  for  finding  the 
hypothenuse'?  —  287.  What  for  finding  the  perpendicular  ?  —  288.  What  for 
finding  the  hase  ? 

54 


278  EVOLUTION. 

2.  Two  vessels  sail  from  tlie  same  port ;  one  sails  due  north 
360  miles,  and  the  other  due  east  450  miles;  what  is  their  dis- 
tance from  each  other  ?  Ans.  576.2-}-  miles. 

3.  The  hypothenuse  of  a  certain  right-angled  ti-iangle  is  60 
feet,  and  the  perpendicular  is  36  feet ;  what  is  the  length  of  the 
base  ?  Ans.  48  feet. 

4.  A  line  drawn  from  the  top  of  the  steeple  of  a  certain  meet- 
ing-house to  a  point  at  the  distance  of  50  feet  on  a  level  from  the 
base  of  the  steeple,  is  120  feet  in  length  ;  what  is  the  hight  of 
the  steeple  ?  Ans.  109.084-  feet. 

5.  The  hight  of  a  tree  on  an  inland  in  a  certain  river  is  1 60 
feet.  The  base  of  the  tree  is  100  ieet  on  a  horizontal  line  from 
the  river,  and  is  elevated  20  feet  above  its  surface.  A  Ime  ex- 
tending from  the  top  of  the  tree  to  the  further  shore  of  the  river 
is  500  feet.     Required  the  width  of  the  river. 

Ans.  366.47-1-  feet. 

6.  On  the  edge  of  a  perpendicular  rock,  whose  base  is  90  feet, 
on  a  level,  from  a  certain  road  that  is  110  feet  wide,  there  is  a 
tower  160  feet  high;  the  length  of  a  line  extending  from  the 
top  of  the  tower  to  a  point  on  the  opposite  side  of  the  road  is  300 
feet.  What  is  the  elevation  of  the  base  of  the  tower  above  the 
road  ?  Ans.  03.6-[-  feet. 

7.  John  Snow's  dwelhng  is  60  rods  north  of  the  meeting- 
house, James  Briggs's  is  80  rods  east  of  the  meeting-house, 
Samuel  Jenkins's  is  70  rods  south,  and  James  Emerson's  90  rods 
west  of  the  meeting-house  ;  how  far  will  Snow  have  to  travel  to 
visit  his  three  neighbors,  and  then  return  home  ? 

Ans.  428.47-}-  rods. 

8.  A  certain  room  is  24  feet  long,  18  feet  wide,  and  12  feet 
high  ;  required  the  distance  from  one  of  the  lower  corners  to  an 
opposite  upper  corner.  Ans.  32.3-|-  feet. 

289.  A  Circle  is  a  plane  figure  bounded  by  a  curved  line,  every 
part  of  which  is  equally  distant  from  a  point  called  the  center. 

The  Circiimfrroncc  or  Periphery  of  a  circle  is  the 

line  which  bounds  it. 

The  Diameter  of  a  circle  is  a  lina  drawn  through 
the  ci'nter,  and  terminated  by  the  circumference ; 
as  AB. 


I 


289.  What  is  a  circle  ?     The  circuinfcrcnco  of  a  circle  1     The  diamctw  1 


APPLICATION  OF  THE  SQUARE  ROOT.        279 

290.  ATI  ciRCLics  are  to  each  otlier  as  the  squares  of  their  di- 
ameters, semi-diameters,  or  circumferences. 

All  similar  TRiANCrLKS  and  other  rectilineal  figures  are 
to  each  other  as  the  squares  of  their  corresponding  sides. 

291,  To  find  the  side,  diameter,  or  circumference  of  a 
surface,  wliicli  is  similar  to  a  given  surface. 

Rule.  —  State  the  question  as  in  Proportion,  and  square  the  given 
sides,  diameters,  or  circumferences,  and  the  square  root  of  the  fourth  term 
of  the  proportion  will  he  the  required  answer. 

202.  To  find  the  area  of  a  surface  which  is  similar  to 
a  given  surface. 

Rule.  —  State  the  question  as  in  Proportion,  and  square  the  given 
sides,  diameters,  or  circumferences,  and  the  fourth  term  of  the  proportion 
is  the  required  answer. 

Examples  for  Practice. 

Ex.  1.  I  have  a  triangular  piece  of  land  containing  65  acres, 
one  side  of  which  is  100  rods  in  length ;  what  is  the  length  of 
the  corresponding  side  of  a  similar  triangle  containing  32J-  acres? 

Ans.  70.71 -{-"rods. 

OPERATION. 

65  :  32^::10  0-:5000;V5000=7  0.7i+  rods. 

2.  I  have  a  board  in  the  form  of  a  triangle  ;  the  length  of  one 
of  its  sides  is  16  feet.  My  neighbor  wishes  to  purchase  one  half 
the  board ;  at  what  distance  from  the  smaller  end  must  it  be  di- 

,vided  parallel  to  the  base  or  larger  end?        Ans.  11.31-|-  feet. 

3.  There  is  a  triangular  piece  of  land,  the  length  of  one  side 
of  wliich  is  11  rods  ;  required  the  length  of  the  corresponding 
side  of  a  similar  triangle  containing  three  times  as  much. 

Ans.  19.05-|-i'ods. 

4.  The  diameter  of  a  circle  is  6  feet,  and  its  area  is  28.3  feet  ; 
what  is  the  diameter  of  a  circle  whose  area  is  42.5  feet  ? 

Ans.  7.35-f  feet. 

5.  If  an '  anchor,  which  weighs  20001b.,  requires  a  cable  3 
inches  in  diameter,  what  should  be  the  diameter  of  the  cable, 
when  the  anchor  weighs  40001b  ?  Ans.  4.24^  inches. 

6.  A  rope  4  inches  in  circumference  will  sustain  a  weight  of 
10001b. ;  what  must  be  the  circumference  of  a  rope  that  will 
sustain  50001b.?  Ans.  8.94-)-  inches, 

290.  "What  ratio  do  circles  have  to  each  other?  —  291.  Tl'.ernlc  for  finding 
the  side,  diameter,  &c.,  of  a  surface  similar  to  a  given  surface  ?  —  292.  The 
rule  for  finding  the  area  of  a  surface  similar  to  a  given  surface  ? 


280  EVOLUTION. 

7.  There  is  a  triangle  containing  72  square  rod?,  and  one  of 
its  fides  measures  12  rods  ;  what  is  the  area  of  a  similar  triano-le 
whose  corresponding  side  measures  8  rods  ?  Aus.  32  rods. 

8.  A  gentleman  has  a  park,  in  the  form  of  a  right-angkd  tri- 
angle, containing  950  square  rods,  the  longest  side  or  hjjiothenuse 
of  which  is  45  rods.  He  wishes  to  lay  out  another  in  the  same 
form,  with  a  liypothenuse  ^  the  length  of  the  first ;  required  the 
area.  Ans.  105.55-|-  square  rods. 

9.  If  a  cylinder  6  inches  in  diameter  contain  1.178-]-  cubic 
feet,  how  many  cubic  feet  will  a  cylinder  of  the  same  length  con- 
tain that  is  9  inches  in  diameter?  Ans.  2.65-)-  i'aet. 

10.  If  a  pipe  2  inches  in  diameter  will  fill  a  cistern  in  20 1 
minutes,  how  long  would  it  take  a  pipe  that  is  o  inches  in  diam- 
eter ?  Ans.  9  minutes. 

11.  A  tube  f  of  an  inch  in  diameter  will  empty  a  cistern  in  50 
minutes  ;  required  the  time  it  will  empty  the  cistern,  when  there 
is  another  pipe  running  into  it  ^  of  an  inch  in  diameter. 

Ans.  62-^^  minutes. 

293.  To  find  the  side  of  a  square  that  can  be  inscribed 
in  a  circle  of  a  given  diameter. 

A  square  is  said  to  be  inscribed  in  a  circle  when 
each  of  its  angles  or  corners  touches  the  circumfer- 
ence. It  may  be  conceived  to  be  composed  of  two 
right-angled  triangles,  the  base  and  perpendicular  of 
each  being  equal,  and  their  liypothenuse  the  diameter 
of  the  circle,  as  seen  in  -the  diagram.     Hence  the 

Rule.  —  Extract  the  square  root  of  half  the  square  of  the  diameter, 
and  it  i.s'  the  side  of  the  inscribed  square. 

Examples  for  Practice. 

1.  TV  hat  is  the  length  of  one  side  of  a  square  that  can  be  in- 
scribed in  a  circle,  whose  diameter  is  12  feet?    Ans.  8.48-)-  ft. 

2.  How  large  a  square  stick  may  be  hewn  fron*  a  round  one 
Avhich  is  30  inches  in  diameter?       Ans.  21.2-)-  iiTClies  square. 

3.  A  has  a  cylinder  of  lignum-vitje,  19|  inches  long  and  1^ 
inches  in  diameter ;  how  large  a  square  ruler  may  be  yjade  from 
it?  Ans.  1.06-]-  inches  square. 

293.  When  is  a  square  said  to  be  inscribcfl  in  a  circle  ?  Of  wlmt  may  the 
inscribed  sciiiarc  bo  conceived  to  bo  conii>osed  1  Wliat  part  of  tlic  circle  i.s 
the  bypotlicniisc  of  the  two  triangles  1  The  rule  fur  lindiu;,'  the  side  of  llio 
inscribed  square  1 


EXTRACTION  OF  THE  CUBE  ROOT.         281 


EXTRACTION  OF  THE  CUBE  ROOT. 


■^Jk         29li     The  Cube  Root  is  the  root  of  a  third  power.     It   is  ?o 

^^  called,  because  the  cube  or  third  power  of  any  number  represents 
the  contents  of  a  cubic  body,  of  which  the  cube  root  is  one  of  its 
sides. 

295.  To  extract  the  cube  root  of  a  number,  is  to  find  such  a 
factor  as,  when  multiplied  into  itself  twice,  will  produce  the  given 
number ;  or  it  is  to  resolve  the  number  into  three  equal  factors. 
Hoots  of  the  first  ten  integers  and  their  cubes  are, 


1, 

2, 

3, 

4, 

5, 

6, 

7, 

8, 

9, 

10 

1, 

8, 

27, 

64, 

125, 

216, 

343, 

512, 

729, 

1000 

It  will  be  observed  that  the  cube  or  third  power  of  each  of  the  num- 
bers contains  three  times  as  many  figures  as  the  root,  or  three  times  as 
many  Avanting  one,  or  two  at  most.  Hence,  to  determine  the  number 
of  figures  in  the  cube  root  of  a  given  number. 

Beginning  at  the  right,  point  it  off  into  as  many  periods  as  possible  q/ 
three  figures  each,  and  there  will  be  as  many  figures  in  the  root  as  there 
are  periods. 

Ex.  1.  I  Lave  17576  cubical  blocks  of  marble,  which  measure 
one  foot  on  each  side  ;  what  will  be  the  length  of  one  of  the 
sides  of  a  cubical  pile  which  may  be  formed  of  them  ? 

Ans.  26  feet. 

OPERATION.  ,  The    number    of 

-i  n  K  n  o  f  nn    r>     4.  blocks  or  feet  on    a 

1  7  5  7  6  (  26,  Root.         ^5^}^.^^^  ^^  ^^^^1  ^^ 

" the     cube     root    of 

2^X300  =  1200)957  6, 1st  dividend.      ^  ^576.  (Art.  2D4.) 

Begmnmg   at   the 

7  2  0  0,  1st  addition,      right  "hand,  we  point 

6'  X  2  X  30        =         2  1  6  0,  2d    addition.      ofi'  the  number  into 

6^      =  2  1  6,  3d  addition,      periods,  by  placing  a 

point  over  the  right- 

9  5  7  6,  Subtrahend,      hand  figure  of  each 

period.  We  then 
find  the  greatest  cube  number  in  the  left-hand  period,  17  (thousands), 
to  be  8  (thousands),  and  its  root  2,  which  we  place  in  the  quotient  or 

294.  "What  is  the  cube  root,  and  why  so  called  ?  —  295.  "What  is  meant  by 
extractintr  the  cube  root?  How  many  more  figures  in  the  cube  of  any 
number  than  in  the  roof?  How  do  you  ascertain  tlie  number  of  fijrures  in 
tlio  cube  root  of  any  number  ?  What  is  found  by  extracting  the  cube  root 
of  tbe  number  in  the  example?  What  is  first  "done  after  separating  the 
number  into  periods  1 

24* 


282 


EVOLUTION. 


root.  As  2  is  in  tlie  place  of  tens,  because  tliere  is  to  be  another  figure 
in  the  root,  its  value  is  20,  and  it  represents  the  side  of  a  cube  (Fig.  1) 
the  contents  of  which  are  80U0  cubic  feet ;  thus,  20  X  20  X  20  =  8000. 

Fig.  1.  ^    We  now  subtract  the  cube  of  2  (tens) 

20  =8  (thousands)  from  the  first  period,  1 7 

(thousands),  and  have  9  (thousand)  feet 
remaining,  which,  being  increased  by  the 
2Q  next  period,  makes  95  76  cubic  feet.  This 
must  be  added  to  three  sides  of  the  cube, 
Fig.  1,  in  order  that  it  may  remain  a  cube. 
To  do  this,  we  must  find  the  supei'ficial 
contents  of  the  three  sides  of  the  cube,  to 
which  the  additions  are  to  be  made.  Now, 
20  since  one  side  is  2  (tens)  or  20  feet  square, 

its  superficial  contents  will  be  20  X  20  = 
400  square  feet,  and  this  multiplied  by  three  will  be  the  superficial  con- 
tents of  three  sides;  thus,  20  X  20  X  3  =  1200,  or,  wliicli  is  the 
same  thing,  we  nmltiply  the  square  of  the  quotient  figure,  or  root,  by 
300;    thus,  2^   X    3o0  =  1200  square  leet.     Making  this  number  a 

divisor,  we  divide  the  dividend  9576  by 
it,  and  obtain  6,  which  we  place  in  the 
root.  This  G  represents  the  thickness  of 
each  of  the  three  additions  to  be  made  to 
the  cube,  and  their  superficial  contenta 
being  multiplied  by  it,  we  have  1200  X  ^ 
=  7200  cubic  feet  for  the  contents  of  the 
three  additions,  A,  B,  and  C,  as  seen  in 
Fig.  2. 

Having  made  these  additions  to  the  cube, 
we  find  tliat  there  are  three  other  defi- 
ciencies, n  n,  0  n,  and  r  r,  the  length  of 
which  is  etpial  to  one  side  of  the  additions, 
2  (tens),  or  20  feet ;  and  their  breadtli  and  thickness,  6  feet,  equal  to 
the  thickness  of  the  additions.  Therefore,  to  find  the  solid  contents  of 
tlic  additions,  necessary  to  supply  these  deficiencies,  we  multiply  the 
product  of  their  length,  breadth,  and  thickness,  by  the  number  of  ad- 
ditions ;  thus,  6X*'X20X3=  21G0,  or,  wliich  is  tlie  same  thing,  we 
multiply  the  square  of  the  last  quotient  figure  liy  the  fci-mer  figure  of 
tlie  root,  and  that  product  by  30  ;  thus,  G- X  2  X  30  =  2160  cubic  feet 
for  the  contents  of  the  additions  s  s,  u  u,  and  v  v.,  as  seen  in  Fig.  3. 


Fig.  2. 


20.5.  Wliat  is  flonc  with  this  greatest  cubo  number,  nnd  what  part  of  Fig.  I 
docs  it  rcprosoiU  ?  Wliat  is  done  witli  tiic  root?  W!i;it  is  its  value,  and 
what  ]iait  of  the  firrtin;  docs  it  re])rescnt  ?  How  are  the  euhieal  contents  of 
tlie  n;;iire  found  1  What  constitutes  tlie  remainder  after  snhtraetint:;  the  cnbo 
number  from  the  left-hand  period  ?  To  how  many  sides  of  the  cube  must 
this  remainder  bo  added?  How  do  you  lind  the  divisor?  What  parts  of 
the  (i<,'uro  does  it  represent?  How  do  you  obtain  the  last  fiirure  of  the  root? 
What  part  of  Fiir.  2  does  it  represent?  What  parts  of  the  figiu-c  docs  tho 
product  represent?     Whiit  throe  other  dcliciciicies  in  the  fij^urc? 


EXTRACTION  OF  THE  CUBE  ROOT. 


283 


These  additions  being  made  to  the  cube, 
■we  still  observe  another  di'licienoy  of  the 
cubical  space  a:  x  a;,  the  length,  breadth,  and 
thickness  of  which  are  each  equal  to  the 
thickness  of  the  other  additions,  or  6  feet.  ^, 
Therefore,  we  find  the  contents  of  the  addi- 
tion necessary  to  supply  this  deficiency  by 
multiplying  its  length,  breadth,  and  thick- 2q 
ness  together,  or  cubing  the  last  figure  of 
the  roof,  thus  6X^X6=  21 6  cubic  feet 
for  the  contents  of  the  ddition-  z  z  s,  as 
seen  in  Fig.  4. 

If  we  add  together  the  several  additions 
that  have  been  made,  thus,  7200  -\-  2160 
_|-  216  =  9576,  we  obtain  the  number  of 
cubic  feet  remaining  after  subtractinaj  the 
first  cube,  which,  being  subtracted  from  the 
dividend  in  the  operation,  leaves  no  re- 
mainder. Hence  the  cubical  pile  formed 
is  26  feet  on  each  side ;  since  26  X  26  X  2635 
=  17576,  the  given  number  of  blocks,  and 
the  sum  of  the  several  parts  of  Fig.  4. 
Thus,  8000 -f  7200 +  2160 +  216  =  17576. 


Rule.  —  Separate  the  given  number  into  as  many  periods  as  possible 
of  three  Jigures  each,  by  placing  a  point  over  the  unit  figure,  and  every 
third  figure  beyond  the  place  of  units. 

Find  the  greatest  cube  in  the  left-hand  period,  and  place  its  root  on  the 
right.  Subtract  the  cube,  thus  found,  from  this  pei'iod,  and  to  the  remain- 
der bring  down  the  next  period  for  a  dividend. 

Mnltipily  the  square  of  the  root  already  found  by  300  for  a  divisor,  by 
which  divide  the  dividend,  and  place  the  quotient,  usually  diminished  by 
one  or  two  units,  for  the  next  figure  of  the  root. 

Multijdy  the  divisor  by  the  last  figure  of  the  root,  and  write  the  product 
under  the  dividend;  then  multiply  the  square  of  the  last  figure  of  the  root 
by  its  former  figure  or  figures,  and  this  product  by  30,  and  place  the 
product  under  the  last;  under  all  set  the  cube  of  the  last  figure  of  the 
root,  and  call  th'sir  sum  the  subtrahend. 


205.  How  do  you  find  their  contents  ?  What  part-;  of  Fig.  3  docs  the 
product  represent  ?  Whnt  other  deficiency  do  you  ol).~ervc  ?  To  whiit  are 
its  length,  breadth,  and  thickness  equal?  How  do  you  find  its  contents'? 
What  part  of  Fig.  4  does  it  represent  1  The  rule  for  extracting  the  cube 
root? 


284  EVOLUTION. 

Subtract  the  subtrahend  from  the  dividend,  and  to  the  remainder  bring 
down  the  next  period  for  a  new  dividend,  with  which  proceed  as  before ; 
and  so  on,  till  the  whole  is  completed. 

Note  1.  —  When  the  number  of*  the  figures  in  the  given  number  is  not 
divisible  exactly  by  3,  the  period  on  the  left  will  contain  less  than  3  figures. 

Note  2., —  The  observations  made  in  Notes  2,  3,  and  4,  under  square 
root,  are  equally  applicable  to  the  cube  rout,  except  in  pointing  otf  decimals 
each  period  must  contain  three  figures,  and  tico  ciphers  must  be  placed  at  the 
risrht  of  the  divisor  when  it  is  not  contained  in  the  dividend. 


'tj' 


Examples  for  Practice. 
1.  What  is  the  cube  root  of  78402752  ?  Ans.  428. 

OPERATION. 

78402752(42  8,  Root. 
64 

4800)14402  =  1st  dividend. 

9600 

480 

8 

10088  =  1  St  subtrahend. 


5  2  9  2  0  0  )4314752  =  2d  dividend. 

4233600 

80640 

512 


4314752  =2d  subtrahend. 

2.  What  is  the  cube  root  of  74088?  Ans.  42. 

3.  What  is  the  cube  root  of  185193  ?  Ans.  57. 

4.  What  is  the  cube  root  of  80621568?  Ans.  432. 

5.  What  is  the  cube  root  of  176558481  ?  Ans.  561. 

6.  What  is  the  cube  root  of  257259456?  Ans.  636. 

7.  Wliat  is  the  cube  root  of  1860867  ?  Ans.  123. 

8.  What  is  the  cube  root  of  1879080904  ?  Ans.  1234. 

9.  What  is  the  cube  root  of  41673648.563  ?  Ans.  346.7. 

10.  What  is  the  cube  root  of  483921.516051  ?  Ans.  78.51. 

11.  What  is  the  cube  root  of  8.144865728  ?  Ans.  2.012. 

12.  What  is  the  cube  root  of  .075686967  ?  Ans.  .423. 

295.  How  many  ciphers  must  be  placed  at  the  right  of  the  divisor  when  it 
is  not  contained  in  the  dividend  7 


^   APPLICATION  OF  THE  CUBE  KOOT.        285 

296.  When  it  is  required  to  extract  the  cube  root  of  a  common 
fraction,  or  a  mixed  number,  it  is  prepared  in  the  same  manner 
as  directed  in  square  root.     (Art.  282.) 

« 

Examples  for  Practice. 

1.  What  is  the  cube  root  of  81y5j^?  Ans.  4.334+ 

2.  What  is  the  cube  root  of  ^-(ys^  ?  -^"S.  ■^^. 

3.  What  is  the  cube  root  of  49^7-  ?  Ans.  3§. 

4.  What  is  the  cube  root  of  166f  ?  Ans.  5|. 

5.  What  is  the  cube  root  of  85  j^^.^  ?  Ans.  4f . 

APPLICATION   OF   THE   CUBE  ROOT. 

297t  The  cube  root  may  be  appUed  in  finding  the  dimensions 
and  contents  of  cubes  and  other  sohds. 

1.  A  carpenter  wishes  to  make  a  cubical  cistern  that  shall  con- 
tain 2744  cubic  feet  of  water ;  what  must  be  the  length  of  one  of 
its  sides?  Ans.  14  feet 

2.  A  farmer  has  a  cubical  box  that  will  hold  400  bushels  of 
grain  ;  what  is  the  depth  of  the  box?  Ans.  7.92-|-  feet. 

3.  There  is  a  cellar,  the  length  of  which  is  18  feet,  the  width 
15  feet,  and  the  depth  10  feet;  what  would  be  the  depth  of 
another  cellar  of  the  same  size,  having  the  length,  width,  and 
depth  equal?  Ans.  13.9 2-{- feet. 

298.  A  Sphere  is  a  solid  bounded  by  one  continued  ccrtivex  sur- 
face, every  part  of  which  is  equally  distant  from  a  point  within, 
called  the  center. 


A 


The  Diameter  of  a  sphere  is  a  straight  line  pars- 
ing through  the  center  and  terminated  by  the  sur- 
face ;  as  A  B.  ^ 


299.    A  Cone  is  a  soHd  having  a  circle  for  its  base,  and  its  top 
terminated  in  a  point,  called  the  vertex. 

296.  How  is  a  common  fraction  or  a  mixed  number  prepared  for  extract- 
ing: the  square  root  ?  —  297.  To  what  may  the  cube  root  be  applied  ? 
—  298.  What  is  a  sphere  ?  The  diameter  of  a  sphere? — 299.  AVhat  is  a 
cone? 


286  EVOLUTION. 


The  Altitude  of  a  cone  is  its  perpendicular  liight,  or 
a  line  di-awn  from  the  vertex  perpendicular  to  the  plane 
of  the  base ;  as  B  C. 

300«  Spheres  are  to  each  other  as  the  cubes  of  their  diameters^ 
or  of  their  circunferences. 

Similar  cones  are  to  each  other  as  the  cubes  of  their  altitudes, 
or  the  diameters  of  their  bases. 

All  SIMILAR  SOLIDS  are  to  each  other  as  the  cubes  of  their  ho- 
mologous or  corresponding  sides,  or  of  their  diameters. 

301.  To  find  the  contents  of  any  solid  which  is  similar 
to  a  given  solid. 

Rule.  —  Stale  the  question  as  in  Proportion,  and  cube  the  given  sides, 
diameters,  altitudes,  or  circumferences,  and  the  fourth  term  of  the  propor- 
tion is  the  required  answer. 

302.  To  find  the  side,  diameter,  circumference,  or  alti- 
tude, of  any  solid,  which  is  similar  to  a  given  solid. 

Rule.  —  State  the  question  as  in  Proportion,  and  cube  the  given  sides, 
diameters,  circumferences,  or  altitudes,  and  the  cube  root  of  the  fourth 
term  of  the  proportion  is  the  required  answer. 

Examples  for  Practice. 

1.  If  a  cone  2  feet  in  bight  contains  456  cubic  feet,  what  are 
the  contents  of  a  similar  cone,  the  altitude  of  which  is  3  feet  ? 

An?.  1539  cubic  feet. 

OPERATION. 

2' :  3'  :  :  4  5  6  :  1  5  3  9 

2.  If  a  cubic  piece  of  metal,  the  side  of  which  is  2  feet,  is 
worth  $0.25,  what  is  another  cubical  piece  of  the  same  kind 
Tvorth,  one  side  of  which  is  12  feet?  Ans.  $1350. 

3.  If  a  ball,  4  inches  in  diameter,  weighs  501b.,  what  is  the 
weight  of  a  ball  6  inches  in  diameter?,  Ans.  lG8.7-j-lb. 


200.  Wliat  is  the  altitmle  of  a  cone?  — 300.  "\Vliat  proportion  do  spheres 
hnvo  to  each  othor  ?  What  proportion  do  cones  have  to  earli  otlier  !  What 
proi-ortion  do  all  simihir  <^olids  liave  to  each  other?  — .^01.  What  is  the  rulo 
for  (indinjj  the  contents  of  a  solid  similar  to  a  iriven  solid  ?  —  302.  The  rulo 
for  finding  the  side,  diameter,  &e..  of  a  solid  similai-  to  a  giveu  solid  1 


ARITHMETICAL   PROGRESSION.  287 

4.  If  a  sujiar-loaf,  which  is  12  inches  in  hight,  weighs  161b., 
how  many  inches  may  be  broken  from  the  base,  that  the  residue 
may  weigh  81b.  ?  Ans.  2.5-|-  in. 

5.  If  an  ox,  that  weighs  8001b.,  girts  6  feet,  what  is  the  weight 
of  an  ox  that  girts  7  feet  ?  Ans.  1270.31b. 

6.  If  a  tree,  that  is  1  foot  in  diameter,  make  1  cord,  how 
many  cords  are  there  in  a  similar  tree,  whose  diameter  is  2 
feet  ?  Ans.  8  cords. 

7.  If  a  bell,  30  inches  high,  weighs  10001b.,  what  is  the  weight 
of  a  bell  40  inches  high  ?  Ans.  2370.31b. 

8.  If  an  apple,  6  inches  in  circumference,  weighs  16  ounces, 
what  is  the  weight  of  an  apple  1 2  inches  in  circumference  ? 

Ans.  128  ounces. 

9.  A  and  B  own  a  stack  of  hay  in  a  conical  form.  It  is  15 
feet  high,  and  A  owns  §  of  the  stack;  it  is  required  to  know 
how  many  feet  he  must  take  from  the  top  of  it  for  his  share. 

Ans.  13.14-  feet. 


ARITHMETICAL    PROGRESSION. 

303t  Arilhmotical  Progression  is  a  series  of  numbers  that  in- 
creases or  decreases  by  a  constant  difference. 

The  Terms  of  the  series  are  numbers  of  which  it  is  formed. 

The  Extremes  are  the  first  and  last  terms. 

The  Means  are  the  terms  between  the  extremes. 

The  Common  Difference  is  the  constant  difference  between  the 
terms. 

The  series  is  ascending  when  each  term  after  the  first  exceeds 
that  bi.'fore  it,  and  descending  when  each  term  after  the  first  is 
less  than  that  before  it. 

Thus,  2,    5,    8, 11,  14,  17,  20,  23,  26,  29,  is  an  ascending  series, 
and    29,  26,  23,  20,  17,  14,  11,    8,    5,    2,  is  a  descending  series. 

303.  ViTiat  is  arithmetical  progression  ?  What  are  the  terms  of  a  pro- 
gression'? The  extremes?  The  means?  Common  difl'creiice ?  What  is 
an  ascending  series  ?     A  descending  series  ? 


288  AKITHMETICAL   PKOGRESSION. 

In  Arithmetical  Progression,  the  first  term,  the  last  term,  the  2 

number  of  terms,  the  common  difference,  and  the  sum  of  the  terms,  .1 

are  so  related  to  each  other,  that  any  three  of  these  being  given,  | 
the  two  others  may  be  readily  determined. 

304.    To  find  the  common  difference,  the  first  term, 
last  term,  and  number  of  terms  being  given. 

Illustkation.  —  In  the  following  series, 

2,  5,  8,  11,  14,  17,  20,  23,  26,  29, 

2  and  29  are  the  extremes,  3  the  common  difference,  10  the  number 
of  terms,  and  the  sum  of  the  series  155. 

It  is  evident  that  the  number  of  common  differences  in  any  series 
must  be  1  less  than  the  number  of  terms.  Therefore,  since  the  number 
of  terms  in  this  series  is  10,  the  number  of  common  differences  will  be 
10 —  1  =  9,  and  their  sum  will  bo  equal  to  the  difference  of  the  ex- 
tremes;  hence,  if  the  difference  of  the  extremes  (29  —  2  =  27)  be 
divided  by  the  number  of  common  differences,  9,  the  quotient,  3,  will 
be  the  common  difference.     Hence  the 

Rule.  —  Divide  the  difference  of  the  extremes  hy  the  number  of  terms 
less  one,  and  ike  quotient  will  be  the  common  difference. 


Examples  for  Practice. 

1.  The  extremes  of  a  series  are  3  and  35,  and  the  number  of 
terms  is  9  ;  what  is  the  common  difference  ?  Ans.  4. 

OPERATION. 

3  5  —  3       , 

^Q ^  =  4,  common  difference. 

2.  If  the  first  term  is  7,  the  last  term  55,  and  the  number  of 
terms  17,  required  the  common  difference.  Ans.  3. 

3.  If  the  first  term  is  4,  the  last  term  14,  and  the  number  of 
terms  15,  vvliat  is  the  common  difference?  Ans.  ^^. 

4.  If  a  man  travels  10  days,  and  the  first  day  goes  9  miles, 
and  the  last  1 7  miles,  and  increases  each  day's  travel  by  an  equal 
difference,  what  is  the  daily  increase  ?  Ans.  |  miles. 

.'W.T.  Wlmt  five  thintrs  arc  namerl,  any  tliroc  of  wliicli  hcins  givon  the  other 
two  can  he  found  ?  —  .'104.  Tlie  rule  for  findinc;  the  commou  diU'creucc,  tho 
first  term,  last  term,  and  number  of  terms  being  given  t 


ARITHMETICAL   rEOGRESSION.  289 

305.  To  find  the  sum  of  all  the  terms,  the  first  term, 
last  term,  and  number  of  terms  being  given. 

Illustration.  —  Let  the  two  following  series  be  arranged  as 
follows :  — 

2,      5,      8,    11,    14,    17,    20,  =    77,  sum  of  first  series. 
20,    17,    14,    11,      8,      5,      2,=    77)  sum  of  inverted  series. 

22,    22,    22,    22,    22,    22,    22,  =  154,  sum  of  both  series. 

From  the  aiTangement  of  the  above  series,  we  see  that,  by 
adding  the  two  as  they  stand,  we  have  the  same  number  for  the 
sum  of  the  successive  terms,  and  that  the  sum  of  both  series  is 
double  the  sum  of  either  series. 

It  is  evident  that,  if  22  in  the  above  series  be  multiplied  by  7, 
the  number  of  terms,  the  product  will  be  the  sum  of  both  series ; 
thus,  22  X  7  =  154 ;  and,  therefore,  the  sum  of  either  series 
will  be  154  -f-  2  =  77.  But  22  is  the  sum  of  the  extremes  in 
each  series ;  thus,  20  -|-  2  =  22.  Therefore,  if  the  sum  of  the 
extremes  be  multiplied  by  the  number  of  terms,  the  product  will 
be  double  the  sum  of  either  series.     Hence,         • 

Rule  1.  —  Multijthj  tJie  sum  of  the  extremes  ty  the  number  of  terms 
and  half  the  product  will  be  the  sum  of  the  series.     Or, 

Rule  2.  —  Multiply  the  sum  of  the  extremes  by  half  the  number  of 
terms,  and  the  product  will  be  the  required  sum. 

Examples  for  Practice. 

1.  If  the  extremes  of  a  series  are  5  and  45,  and  the  number 
of  terms  9,  what  is  the  sum  of  the  series  ?  Ana.  225. 

OPEKATION. 

(45  +  5)  X  9 

=225,  sum  of  the  series. 

2.  John  Oaks  engaged  to  labor  for  me  12  months.  For  the 
first  month  I  was  to  pay  him  $  7,  and  for  the  last  month  $  51. 
In  each  successive  month  he  w^as  to  have  an  equal  addition  to 
his  wages ;  what  sum  did  he  receive  for  his  year's  labor  ? 

Ans.  $  348. 


305.  The  rule  for  finding  the  sum  of  all  the  terms,  the  first  term,  last  term, 
and  number  of  terms  being  given  1 
25 


290  ARITHMETICAL  PROGRESSION. 

3.  I  have  purchased  from  "W.  Hall's  nursery  100  fruit-trees 
of  various  kinds,  to  be  set  around  a  circular  lot  of  land  at  the 
distance  of  one  rod  from  each  other.  Having  deposited  them  on 
one  side  of  the  lot,  how  far  shall  I  have  traveled  when  I  have 
set  out  my  last  tree,  provided  I  take  only  one  tree  at  a  time,  and 
travel  on  the  same  line  each  way  ?  Ans,  9801  rods. 

306.  To  find  the  number  of  terms,  the  extremes  and 
common  difference  being  given. 

Illustration.  —  Let  the  extremes  of  a  series  be  2  and  29, 
and  the  common  difference  3.  The  difference  of  the  extremes 
will  be  29  —  2  =  27.  Now,  it  is  evident  that,  if  the  difference  of 
the  extremes  be  divided  by  the  common  difference,  the  quotient 
will  be  the  number  of  common  differences ;  thus,  27  -r-  3  =  9. 
It  has  been  shown  (Art.  304)  that  the  number  of  terms  is  1  more 
than  the  number  of  differences  ;  therefore  9  -)-  1  =^  10  is  the 
number  of  terms  in  this  series.     Hence  the 

EuLE.  —  Divide  the  difference  of  the  extremes  by  the  common  differ- 
ence,  and  the  quotient,  increased  by  1,  will  be  the  number  of  terms. 

Examples  for  Practice. 

1.  If  the  extremes  of  a  series  are  4  and  44,  and  the  common 
difference  5,  what  is  the  number  of  terms  ?  Ans.  9. 

OPEKATIONi 

44  —  4  , 

-f-  1  =  9,  number  of  terms. 

2.  A  man  going  a  journey  traveled  the  first  day  8  miles,  and 
the  last  day  47  miles,  and  each  day  increased  his  journey  by  3 
miles.     How  many  days  did  he  travel?  Ans.  14  days. 

307.  To  find  the  sum  of  the  terms,  the  extremes  and 
common  difference  being  given. 

Illustration.  —  Let  the  extremes  be  2  and  29,  and  the  com- 
mon difference  3.  The  difference  of  the  extremes  will  be  29  — • 
2  =  27  ;  and  it  has  been  shown  (Art.  306)  that  if  the  dill'er- 
ence  of  the  extremes  be  divided  by  the  common  difference,  the 

306.  The  rule  for  finding  the  number  of  terms,  tho  extremes  and  common 
diirercnce  being  given  ? 


ARITPIMETICAL  PROGRESSION.  291 

quotient  will  be  the  number  of  terms  less  one.  Therefore,  the 
number  of  terms  less  one  will  be  27  -4-  3  =  9,  and  the  number 
of  terms  9  -|-  1  =  10.  It  was  also  shown  (Art.  305)  that,  if  the 
number  of  terms  be  multiplied  by  the  sum  of  the  extremes,  and 
the  product  divided  by  2,  the  quotient  will  be  the  sum  of  the 
terms.     Hence  the 

Rule.  —  Divide  the  difference  of  the  extremes  hj  the  common  differ' 
ence,  and  to  the  quotient  add  1  ;  midtiplij  this  sum  by  the  sum  of  the 
extremes,  and  half  the  product  is  the  sum  of  the  series. 

Examples  for  Practice. 

1.  If  the  two  extremes  are  11  and  74,  and  the  common  differ- 
ence 7,  what  is  the  sum  of  the  series  ?  Ans.  425. 

OPERATION. 

74—11,,       ,-(74  +  ll)XlO       .^.  ■         . 

H  1  =  1  0  :  ^ ■ — ^        =425,  sum  of  series. 

7         '  2 

2.  A  pupil  conunenced  Virgil  by  reading  12  lines  the  first  day, 
17  lines  the  second  day,  and  thus  increased  eveiy  day  by  5  lines, 
until  he  read  137  hues  in  a  day.  How  many  lines  did  he  read  m 
all?  Ans.  1937  lines. 

308.  To  find  the  last  term,  the  first  term,  the  number 
of  terms,  and  the  common  difference  being  given. 

Illustration.  —  Let  the  first  term  of  a  series  be  2,  the  num- 
ber of  terms  10,  and  the  common  difference  3.  It  has  been  shown 
(Ai't.  304)  that  the  number  of  common  differences  is  always  1 
less  than  the  number  of  terms  ;  and  that  the  sum  of  the  common 
differences  is  equal  to  the  difference  of  the  extremes  ;  therefore, 
since  the  number  of  terms  is  10,  and  the  common  difference  3,  the 
difference  of  the  extremes  will  be  (10 —  1)  X  3  =  27  ;  and  this 
difference,  added  to  the  first  term,  must  give  the  last  term ;  thus, 
2  +  27  =  29.     Hence  the 

Rule.  —  Multiply  the  number  of  terms  less  1  by  the  common  differ- 
ence, and  add  this  product  to  the  first  term  for  the  last  term. 

Note.  —  If  the  series  is  descending,  the  product  must  be  s«6<racterf  from 
the  first  term. 

307.  The  rule  for  finding  the  sum  of  the  series,  the  extremes  and  common 
difterence  being  given  ?  — 308.  The  rule  for  finding  the  last  term,  the  first 
term,  the  number  of  terms,  and  common  difference  being  given  ? 


292  ANNUITIES   AT   SIMPLE   INTEREST. 

Examples  for  Practice. 

1.  If  the  fii'st  term  is  1,  the  number  of  terms  7,  and  the  com- 
mon difference  6,  what  is  the  last  term  ?  Ans.  37. 

OPEKATION. 

l4-(7  — 1)X6  =  3  7,  lastterm. 

2.  If  a  man  travel  7  miles  the  first  day  of  his  journey,  and  9 
miles  the  second,  and  shall  each  day  travel  2  miles  farther  than 
the  preceding,  how  far  will  he  travel  the  twelfth  day  ? 

Ans.  29  miles. 

3.  If  A  set  out  from  Portland  for  Boston,  and  travel  20|^  miles 
the  first  day,  and  on  each  succeeding  day  1^  miles  less  than  on 
the  preceding,  how  far  will  he  travel  the  tenth  day  ? 

Ans.  6f  miles. 
« 
ANNUITIES    AT    SIMPLE    INTEREST. 

309.  An  Annuity  is  a  sum  of  money  to  be  paid  annually,  or  at 
any  other  regular  period,  either  for  a  limited  time  or  forever. 

■   The  Present  Worth  of  an  annuity  is  that  sum  which  being  put  at 
interest  will  be  sufficient  to  pay  the  annuity. 

The  Amount  of  an  annuity  is  the  interest  of  all  the  payments 
added  to  their  sum. 

Annuities  are  said  to  be  in  arrears  when  they  remaiu  unpaid 
after  they  have  become  due. 

310.  To  find  the  amount  of  an  annuity  at  simple  in- 
terest. 

Ex.  1.  A  man  purchased  a  farm  for  $2000,  and  agreed  to  pay 
for  it  in  5  years,  paying  $400  annually  ;  but,  finding  himself  un- 
able to  make  tlie  annual  payments,  he  agreed  to  pay  the  whole 
amount  at  the  end  of  the  5  years,  with  the  simple  interest,  at  6 
per  cent.,  on  each  payment,  from  the  time  it  became  due  till  the 
time  of  settlement ;  what  did  the  farm  co=>t  him  ?    Ans.  %  2240. 

Illustration.  —  It  is  evident  the  ffth  payment  will  be 
$  400,  without  interest ;  the  foitrth  will  be  on  interest  1  year,  and 
will  amount  to  $  424  ;  the  third  will  be  on  interest  2  years,  and 
will  amount  to  $  448  ;*  the  second  will  be  on  interest  3  years, 

309.  What  is  an  annuity?  What  is  meant  by  the  present  worth  of  an 
annuity  ?     By  the  amount  J     When  arc  annuities  said  lo  be  iu  arrears  ' 


ANNUITIES   AT   SIMPLE   INTEREST.  293 

and  will  amount  to  $  472 ;  and  the  Jirst  will  be  on  interest  4 
years,  and  will  amount  to  $  49 G.  Therefore,  these  several  sums 
form  an  arithmetical  series  ;  thus  400,  424,  448,  472,  496  ;  of 
which  the  fifth  payment,  or  the  annuity,  is  the  first  term,  the  in- 
terest on  the  annuity  for  one  year  the  common  difference,  the  time 
in  years,  the  number  of  terms,  and  the  amount  of  the  annuity,  the 
sum  of  the  series.     The  sum  of  this  series  is  found  by  Art.  305  ; 

thus,  (i5^+i55IXi  =  12210.    IWthe 

Rule.  —  First  find  the  last  term  of  the  series  (Art.  308),  and  then  the 
sum  of  the  to-rns  (Art.  305). 

Note.  —  If  the  payments  are  to  be  made  semi-annually,  quarterly,  &c., 
these  periods  will  be  the  number  of  terms,  and  the  interest  of  the  annuity  for 
each  period  the  common  diiierence. 

Examples  for  Practice. 

2.  What  will  an  annuity  of  $  250  amount  to  in  6  years,  at  6 
per  cent,  simple  interest  ?  Ans.  $  1725. 

3.  "What  will  an  annuity  of  $  380  amount  to  in  10  years,  at  5 
per  cent,  simple  interest?  Ans.  $4655. 

4.  An  annuity  of  $  825  was  settled  on  a  gentleman,  January  1, 
1840,  to  be  paid  annually.  It  was  not  paid  until  January  1, 
1848  ;  how  much  did  he  receive,  allowing  6  per  cent,  simple  in- 
terest ?  Ans.  $  7986. 

5.  A  gentleman  let  a  house  for  3  years,  at  $  200  a  year,  the 
rent  to  be  paid  semi-annually,  at  8  per  cent,  per  annum,  simple 
interest.  The  rent,  however,  remained  unpaid  until  the  end  of 
the  three  years  ;  what  did  he  then  receive  ?  Ans.  $  660. 

6.  A  certain  clergyman  was  to  receive  a  salary  of  $  700,  to  be 
paid  annually ;  but  his  parishioners  neglected  to  pay  him  for  8 
years  ;  but  he  agreed  to  settle  with  them,  and  allow  them  $  100 
if  they  would  pay  him  liis  just  due  with  interest ;  required  the 
sum  received.  Ans.  $  6676. 

7.  A  certain  gentleman  in  Boston  has  a  very  fine  house,  which 
he  rents  at  $  50  per  month.  Now,  if  his  tenant  shall  omit  pay- 
ment until  the  end  of  the  year,  what  sum  should  the  owner  re- 
ceive, reckoning  interest  at  12  per  cent.  ?  Ans.  $  633. 

310.  What  forms  the  first  tcrrn  of  a  progression  in  an  annuity  ?  What 
the  common  diiTerence  ?  The  number  of  terms  1  The  sum  of  the  series  ? 
The  rule  for  finding  the  amount  of  an  annuity  at  simple  interest  1  If  the 
payments  are  made  semi-annually,  quarterly,  &c.,  what  constitute  the  terms? 
What  the  common  difference  1 
25* 


294  GEOMETRICAL   PROGRESSION. 

GEOMETRICAL    PROGRESSION. 

Silt  Geometrical  Progression  is  a  series  of  numbers  increasing 
by  a  constant  multiplier,  or  decreasing  by  a  constant  divisor. 

•  The  Ratio  is  the  constant  multipUer  or  divisor. 

The  Terms  are  the  numbers  of  which  the  series  is  formed. 

The  Extremes  are  the  first  and  last  terms. 

Tlie  Means  are  the  terms  between  the  extremes. 

The  series  is  ascending  when  each  term  after  the  first  increases 
by  a  constant  multiplier,  and  descending  when  each  term  after  the 
first  decreases  by  a  constant  diA'isor. 

Thus:    2,      4,      8,    16,    32,    64,  is  an  ascending  series, 
and       64,    32,    16,      8,      4,      2,  is  a  descending  series. 

In  the  first  series  the  constant  multiplier,  2,  is  the  ratio,  and  in 
the  second,  the  constant  divisor,  2,  is  the  ratio. 

In  a  geometrical  series  the  product  of  the  extremes  is  equal  to 
the  product  of  any  two  of  the  means  equally  distant  from  the 
extremes.  Thus,  in  the  above  series,  2X64=4X32  =  8X 
16  =  128. 

In  Geometrical  Progression  the  Jii'c  parts  are  so  related  to 
each  other,  that  any  three  of  the  following  being  given,  the  two 
others  may  be  readily  determined:  — 

1st.   The  first  term  ; 

2d.    The  last  term  ; 

3d.    The  number  of  terms ; 

4th.  The  ratio ; 

5th.  The  sum  of  the  terms. 

312.   To  find  a  required  extreme,  when  the  other  ex-. 
tremc,  the  ratio,  and  the  number  of  terms  are  given. 

Illustration.  —  Let  the  first  term  be  2,  the  ratio  3,  and  the 
number  of  terms  7.  It  is  evident  that,  if  we  multiply  the  jirst 
term  by  the  ratio,  the  product  will  be  the  second  term  in  the 
series ;  and  if  we  will  multiply  the  second  term  by  the  ratio,  the 
product  will  be  the  third  term  ;  and  so  on.  The  seventh,  or  last 
term,  therefore,  must  be  the  result  of  six  such  multiplications  ;  or 
the  product  of  the  first  term,  2,  by  3*,  or  2  X  729  =  1458. 

311.  What  is  geometrical  projrrcssion ?  What  is  the  ratio?  Wliat  arc 
the  extremes  of  a  scries?  Tlic  means?  When  is  a  series  ascendinsr? 
When  flcseendinp  ?  What  five  things  are  mentioned,  any  three  of  which 
being  given,  tlie  other  two  may  l>c  found  1 


Geometrical  progression.  295 

If  the  last  term  had  been  given  and  the  first  required,  the 
process  would  evidently  have  been  by  division,  since  every  less 
term  is  the  result  of  a  division  of  the  term  next  larger,  by  ratio. 
Hence  the 

Rule.  —  Raise  the  ratio  to  a  power  wJiose  index  is  equal  to  tJie  number 
of  terms  less  one;  then  multiply  this  power  by  the  first  term,  to  find  the 
last,  or  divide  it  hy  the  last  term  to  find  the  first. 

Note.  —  This  rule  may  be  applied  in  computing  compound  interest, 
the  principal  being  the  first  term,  the  amount  of  one  dollar  for  one  year 
the  ratio,  the  time,  in  years,  one  less  than  the  number  of  terms,  and  the 
amount  the  last  term. 

Examples  for  Practice. 

1.  The  first  term  of  a  series  is  1458,  the  number  of  terms  7, 
and  the  ratio  ^  ;  what  is  the  last  term  ?  Ans.  2. 

OPERATION. 

Eatio  {\y  =  727?  Y^-g-  X  1  4  5  8  =  J/sr-  =  2,  the  last  term. 

2.  If  the  first  term  of  a  series  is  4,  the  ratio  5,  and  the  num- 
ber of  terms  7,  what  is  the  last  term  ?  Ans.  62500. 

3.  If  the  first  term  of  a  series  is  28672,  the  ratio  ^,  and  the 
number  of  terms  7,  what  is  the  last  term  ?  Ans.  7. 

4.  The  first  term  of  a  series  is  5,  the  ratio  4,  and  the  number 
of  terms  is  8  ;  required  the  last  term.  Ans.  81920. 

5.  If  the  first  term  of  a  series  is  10,  the  ratio  20,  and  the  num- 
ber of  terms  5,  what  is  the  last  term.''  Ans.  1600000. 

6.  If  the  first  term  of  a  series  is  30,  the  ratio  1.06,  and  the 
number  of  terms  6,  what  is  the  last  term  ? 

Ans.  40.146767328. 

7.  What  is  the  amount  of  $  1728  for  5  years,  at  6  per  cent., 
compound  mterest.?  Ans.  $  2312.453798+- 

8.  What  is  the  amount  of  $  328.90  for  4  years,  at  5  per  cent., 
compound  interest  ?  Ans.  $  399.78-{-. 

9.  A  gentleman  purchased  a  lot  of  land  containing  15  acres, 
agreeing  to  pay  for  the  whole  what  the  last  acre  would  come 

312.  What  is  the  rule  for  finding  a  required  extreme,  when  one  of  the  ex- 
tremes, the  ratio,  and  number  of  terras  are  given  ?  To  what  may  this  rule 
be  applied  1 


296  GEOMETRICAL   PROGRESSION. 

to,  reckoning  5  cents  for  the  first  acre,  15  cents  for  the  second, 
and  so  on,  in  a  threefold  ratio.     What  did  the  lot  cost  him  ? 

Ans.  $  239148.45. 

313.   To  find  the  suiM  of  all  the  terms,  the  first  term, 
the  ratio,  and  the  number  of  terms  being  given. 

Illustration.  —  Let  it  be  required  to  find  the  sum  of  the 
following  series : 

2,         6,         18,         54. 


o 


If  we  multiply  each  term  of  this  series  by  the  ratio  3,  the 
products  will  be  6,  18,  54,  162,  forming  a  second  series,  whose 
sum  is  three  times  the  sum  of  the  first  series ;  and  the  difference 
between  these  two  series  is  twice  the  sum  of  the  first  series. 
Thus, 

6,     18,     54,     162,  the  second  series. 
2,     6,     18,     54,  the  first  series. 

2,     0,       0,      0,     162  —  2  =  160,  difference  of  the  two  series. 

Now,  since  this  difference  is  twice  the  sum  of  the  first  series, 
one  half  this  difference  will  be  the  sum  of  the  fii'st  series ;  thus 
160 -=-2  =80. 

It  will  be  observed,  that  if  we  had  multiplied  54,  the  last 
term  of  the  first  series,  by  the  ratio  3,  and  from  the  product,  1 62, 
subtracted  2,  the  first  term,  we  should  have  obtained  1 60  ;  and 
this  being  divided  by  the  ratio  3,  less  1,  would  have  given  80  for 
the  sum  of  the  first  sei'ies,  as  before.     Hence  the 

Rule.  —  Find  the  last  term  as  in  Art.  312.  Multiply  hj  the  ratio, 
and  from  the  product  subtract  the  first  term.  Then  dicide  this  remainder 
iy  the  ratio  less  1 ,  and  the  quotient  will  be  the  sum  of  the  series. 

Note  1. —  If  the  ratio  is  loss  than  1,  the  product  of  the  last  term,  multi- 
plied by  the  ratio,  must  be  subtracted  from  the  first  term ;  and,  to  obtain  tho 
divisor,  the  ratio  must  be  subtracted  from  unity,  or  1. 

Note  2.  — When  a  descending  scries  is  continued  to  infinity,  it  becomes 
what  is  called  an  infinite  sekies,  whose  last  term  must  be  regarded  lis  0, 
and  its  ratio  as  a  fraction.     To  find  the  sura  of  an  infinite  series,  — 


313.  The  rule  for  finding  the  sum  of  all  the  terms,  the  first  term,  ratio, 
and  number  of  terms  being  given  ?  If  tho  ratio  is  less  than  a  unit,  what 
must  be  done  with  tiie  product  of  the  last  term  multiplied  by  tho  ratio  i  How 
ia  tlio  divisor  obtained  when  the  ratio  is  less  than  1 1 


GEOMETRICAL   PROGRESSION.  297 

Divide  the  first  term  Inj  1,  decreased  bi/  the  fraction  denoting  the  ratio,  and  the 
qwtieiU  will  be  the  sum  required. 


Examples  for  Practice. 

1.  If  the  first  term  of  a  series  is  12,  the  ratio  3,  and  the  num- 
ber of  terms  8,  what  is  the  sum  of  the  series.  Ans.  39360. 

^  OPERATION. 

Ratio  3^  X  12  =  26244,  the  last  term ;  26244  X  3  =  78732  ; 
78732  —  12  =  78720 ;  78720  h-  (3  —  1)  ==  39360,  the  sum  of 
the  series. 

2.  The  first  term  of  a  series  is  5,  the  ratio  §,  and  the  number 
of  terms  6;  requii'ed  the  sum  of  the  series.  Ans.  13^f|. 

OPERATION. 

Ratio   (f )5  X  5  =  H^,  the   last   term  ;    ^fa  ^  §  =  f  f f  ;    5 

—  m  =  ¥?¥  ;  %¥-  -^  (1  —  f )  =  Wi'  =  13^11,  the  sum  of 
the  series. 

3.  If  the  first  term  of  a  series  is  8,  the  ratio  4,  and  the  num- 
ber of  terms  7,  required  the  sum  of  the  series.         Ans.  43688. 

4.  If  the  first  term  is  10,  the  ratio  f ,  and  the  number  of  terms 
5,  what  is  the  sum  of  the  series  ?  Ans.  oO^^^. 

5.  If  the  first  term  is  18,  the  ratio  1.06,  and  the  number  of 
terms  4,  what  is  the  sum  of  the  sei'ies?  Ans.  78.743-(— 

6.  Wlien  the  first  term  is  S  144,  the  ratio  $  1.05,  and  the  num- 
ber of  terms  5,  what  is  the  sum  of  the  series  ? 

Ans.  $  795.6909. 

7.  D.  Baldwin  agreed  to  labor  for  E.  Thayer  for  6  months. 
For  the  first  month  he  was  to  receive  $  3,  and  each  succeeding 
month's  wages  were  to  be  increased  by  f  of  his  wages  for  the 
month  next  preceding ;  required  the  sum  he  received  for  his  6 
months'  labor.  ■  Ans.  $91|^|-. 

8.  If  the  first  term  of  a  series  is  2,  the  ratio  6,  and  the  number 
of  terms  4,  what  is  the  sum  of  the  series?  Ans.  518. 

9.  A  lady,  wishing  to  purchase  10  yards  of  silk  for  a  new 
dress,  thought  $  1.00  per  yard  too  high  a  price ;  she,  however, 
agreed  to  give  1  cent  for  the  first  yard,  4  for  the  second,  16  for 
the  third,  and  so  on,  in  a  fourfold  ratio ;  what  was  the  cost  of 
the  dress  ?  Ans.  $  3495.25. 


298 


GEOMETRICAL   PROGRESSION. 


ANNUITIES  AT   COMPOUND  INTEREST. 

3U.  An  Annuity  is  at  Compound  Interest  when  compound  intei> 

est  is  reckoned  on  the  annuity  in  arrears. 

The  several  payments  form  a  geometrical  series,  of  which  the 
annuity  is  the  first  term,  the  amount  of  $  1.00  for  one  year  the 
ratio,  the  years  the  number  of  terms,  and  the  amount  of  the 
annuity  the  sum  of  the  series.  • 

315.  To  find  the  amount  of  an  annuity  at  compound 
interest. 

EuLE  1.  —  Find  the  sum  of  the  series,  as  in  Art.  313.     Or, 

Rule  2.  —  Mtdtiply  the  amount  of  $1.00,  for  the  given  time  and 
rate  found  in  the  table,  hy  the  annuity,  and  the  j^roduct  will  be  the  re- 
quired  amount. 

TABLE, 

Showlsg  the  Ajiount  of  $  1  Annuity  at  Cosipound  I^TEEEST,  from 

1  Year  to  40. 


Years. 

6  per  cent. 

6  per  cent. 

Years. 

6  per  cent. 

6  per  cent. 

1 

1.000000 

1.000000 

21 

35.719252 

39.992727 

2 

2.0.50000 

2.060000 

22 

38.505214 

43.392290 

3 

3.152.500 

3.183600 

23 

41.4304T5 

46.995828 

4 

4.310125 

4.374616 

24 

44.501999 

50.815577 

5 

5.525631 

5.637093 

25 

47.727099 

54.864512 

6 

6.801913 

6.975319 

26 

51.113454 

59156383 

7 

8.142008 

8.393838 

27 

54.669126 

63.705766 

8 

9.549109 

9.897468 

28 

58.402583 

68.528112 

9 

11.026564 

11.491316 

29 

62.322712 

73.639798 

10 

12.577893 

13.180795 

30 

66.438847 

79.058186 

11 

14.206787 

14.971643 

31 

70.760790 

84.801677 

12 

15.917127 

16.869941 

»  32 

75.298829 

90.889778 

13 

17.712983 

18.882138 

33 

80.063771 

97.343165 

14 

19.598632 

21.015066 

34 

85.066959 

104.183755 

15 

21.578564 

23.275970 

35 

90.220307 

111.434780 

16 

23.657492 

25.672528 

36 

95.836323 

119.120867 

17 

25.840366 

28.212880 

37 

101.628139 

127.268119 

18 

28.132385 

30.905653 

38 

107.709546 

135.904206 

19 

30.539004 

33.759992 

39 

114.09.5023 

145.058458 

20 

33.065954 

36.785591 

40 

120.799774 

1.54.761966 

314.  When  is  an  annuity  said  to  be  at  componnd  interest?  What  do  the 
amounts  of  the  several  payments  form  ''  What  i.s  the  first  term  of  (ho  series  ? 
The  ratio"?  The  number  of  terms?  The  sum  of  the  series  ?  —  .■',15.  Tho 
first  rule  for  finding  tho  amount  of  an  annuity  \  Tiie  second  \  Wliat  docs 
Clio  table  thow  ? 


•     ANNUITIES   AT   COMPOUND   INTEREST.  299 

Examples  for  Practice. 

1.  "What  will  an  annuity  of  $  378  amount  to  in  5  years,  at  6 
per  cent,  compound  interest?  Aus.  $  2130.821-|-. 

OPERATION   BY   EULE   FIRST. 

1.0  6^  — 


:  J  X  3  7  8  =  $  2  1  3  0.8  2  1+. 


♦  1.0  6  - 

OPERATIOK   BY   RULE   SECOND. 

5.6  37093X378  =  $213  0.8  2  1+. 

2.  "What  will  an  annuity  of  $  1728  amount  to  in  4  years,  at  5 
per  cent,  compound  interest?  Ans.  $7447.896-|-. 

3.  "What  will  an  annuity  of  $  87  amount  to  in  7  years,  at  6 
per  cent,  compound  interest  ?  Ans.  $  730.263-(-. 

4.  What  will  an  annuity  of  $  500  amount  to  in  6  years,  at  6 
per  cent,  compound  interest?  Ans.  $3487.659-(-. 

5.  What  will  an  annuity  of  $  96  amount  to  in  10  years,  at  6 
per  cent,  compound  interest?  Ans.  $  1265.356-]-. 

6.  What  will  an  annuity  of  $  1000  amount  to  in  3  years,  at  6 
per  cent,  compound  interest  ?  Ans.  $  3183.60. 

7.  July  4,  1842,  H.  Piper  deposited  in  an  annuity  office,  for 
his  daughter,  the  sum  of  $56,  and  continued  his  deposits  each 
year,  making  the  last  July  4,  1848.  Required  the  sum  in  the 
office  July  4,  1848,  allowing  6  per  cent,  compound  interest. 

Ans.  $  470.05 4-f-. 

8.  C.  Greenleaf  has  two  sons,  Samuel  and  William.  On  Sam* 
uel's  birthday,  when  he  was  15  years  old,  he  deposited  for  him, 
in  an  annuity  office,  which  paid  5  per  cent,  compound  interest, 
the  sum  of  $  25,  and  this  he  continued  yearly,  making,  however, 
the  last  deposit  on  his  becoming  21  years  of  age.  On  William's 
becoming  12  years  old,  he  deposited  for  him,  in  an  office  which 
paid  6  per  cent,  compound  interest,  the  sum  of  $  20,  and  contin- 
ued this  yearly,  making  the  last  deposit  on  his  becoming  21  years 
of  age.  Wluch  will  receive  the  larger  sum,  when  21  years  of 
age?        Ans.  $  60.0654-  William  receives  more  than  Samuel. 

9.  I  gave  my  daughter  Lydia  $  10  on  her  becoming  8  years 
old,  and  the  same  sum  on  her  birthday  each  year,  giving  the  la? 
on  her  becoming  21  years  old.     This  sum  was  deposited^ 
savings  bank,  which  pays  5  per  cent,  annually.     Rep*" 
amount  in  the  bank  for  her  when  she  is  21  years  of-^ ' 

Ap- 


300  ALLIGATION. 


ALLIGATION. 

316.  Alligation  is  a  process  employed  in  the  solution  of  ques- 
tions relating  to  the  compounding  or  mixing  of  articles  of  differ- 
ent qualities  or  values. 

It  is  of  two  kinds :  AUigation  Medial  and  Alligation  Alternate. 

ALLIGATION  MEDIAL. 

317.  Alligation  Medial  is  the  process  of  finding  the  mean  or 
average  rate  of  a  mixture  composed  of  articles  of  different  quaU- 
ties  or  values,  the  quantity  and  rate  of  each  being  given. 

318.  To  find  the  average  value  of  several'  articles 
mixed,  the  quantity  and  rate  being  given. 

Rule.  —  Find  the  value  of  each  of  the  articles,  and  divide  the  sum 
of  their  values  lij  the  sum  of  the  (juantlties  of  the  articles.  The  quotient 
will  be  the  average  value  of  the  mixture. 

Examples  for  Practice. 

Ex.  1.  A  grocer  mixed  201b.  of  tea  worth  $  0.50  a  poiind,  with 
301b.  worth  $  0.75  a  pound,  and  501b.  worth  $  0.45  a  pound ; 
what  is  1  pound  of  the  mixtui'e  worth  ?  Ans.  $  0.55. 

OPEKATION. 

$  0.5  0  X  2  0  =  S  1  0.0  0  Proof. 

!.^*^^  ^  f  A^f  oo'r  A  $0.5  5X     20  =  $11.00 

^0.4  5  X  50  =  $2  2.50  $  0.5  5  X      3  0  =  $  1  6.5  0 

10  0)  $55.00  $0.55X     50  =  $2  7.50 

$  0.5  5  $  0.5  5  X  1  0  0  =  $  5  5.0  0 

201b.,  at  50  cts.  per  lb.,  is  worth  S  10.00  ;  30lb.,  at  75  cts.  per  lb.,  is 
worth  S  22.50  ;  and  50lb.,  at  45  cts.  per  lb.,  is  worth  S  22.50.  Then, 
201b.  -|-  301b.  4-  501b.  =  lOOlb.,  is  worth  S  10.00  +  S  22.50  _-|-  $  22.50 

S  55.00 ;  and  lib.  is  worth  as  many  dollars  as  100  is  contained  times 
in  55.00,  or  $0.55. 

2.  I  have  four  kinds  of  molasses,  and  a  different  quantity  o! 
each,  as  follows :  30  gal.  at  20  cents,  40  gal.  at  25  cents,  70  gal. 
at  30  cents,  and  80  gal.  at  40  cents ;  what  is  a  gallon  of  the  mix- 
ture worth?  An.s.  $0.31^. 

3.  A  fanner  mixed  4  bush,  of  oats  at  40  cents,  8  bush,  of  corn 

.T16.  What  is  allipation ?  What  two  kinds  arc  there? — 317.  What  is 
alli^'^ation  nuulial  1  —  .318.  Tho  rule  for  fiiulinp:  the  mean  value  of  several 
artielcs  at  dilFcrent  rates  ?  IIow  docs  it  ajjpear  that  this  process  will  give 
Uic  mean  value  of  a  mi.xturc  .' 


ALLIGATION  ALTERNATE. 


301 


at  85  cents,  12  bush,  rye  at  $1.00,  and  10  bush,  of  wheat  at 
$  1.50  per  bushel.  What  Avill  one  bushel  of  the  mixtui-e  be 
worth  ?  Ans. 


1.04^V 


ALLIGATION  ALTERNATE. 


319.  Alligation  Allernate  is  a  process  of  finding  what  quantity 
of  articles,  whose  rate  or  qualities  ai*e  given,  must  be  taken,  to 
compose  a  mixture  of  any  given  rate  or  quality. 

320.  To  find  what  quantity  of  each .  article  must  be 
taken  to  form  a  mixture  of  a  given  rate. 

Ex.  1.  I  wish  to  mix  spice,  at  20  cents,  23  cents,  26  cents,  and 
28  cents  per  pound,  so  that  the  mixture  may  be  worth  25  cents 
per  pound.     How  many  pounds  of  each  must  I  take  ? 


25cts. 


1 

lib. 
lib. 
lib. 
lib. 

'IRST   OPERATION. 

at  20cts.  gain  5cts. 
at  23cts.  gain  2cts. 
at  26cts.  loss  let. 
at  28cts.  loss  Sets. 

at  28cts.  loss  Sets. 

Ans.  - 

riib. 

lib. 

lib. 

_2lb. 

PROOF. 

at  20cts.  =  20cts. 
at  23cts.  =  23cts. 
at  26cts.  ==  26cts. 
at  28cts.  =  5 Gets. 

lib. 

$1 

51b. 
.25-^ 

whole  val.  $1.25 
-  5  =  25cts.  per  lb. 

Compared  with  the  mean  or  average  price  given,  by  taking  lib.  at  20 
cents  there  is  a  gain  of  5  cents,  by  taking  lib.  at  23  cents  a  gain  of  2 
cents,  by  taking  lib.  at  26  cents  a  loss  of  1  cent,  and  by  taking  lib.  at 
28  cents  a  loss  of  3  cents  ;  making  an  excess  of  gain  over  loss  of  3 
cents.  Now,  it  is  evident  that  the  mixture,  to  be  of  the  average  rate 
named,  should  have  the  several  items  of  gain  and  loss  in  the  aggregate 
exactly  offset  one  another.  This  balance  we  can  effect,  in  the  present 
case,  either  by  taking  3lb.  more  of  the  spice  at  26  cents,  or  lib.  more 
of  spice  at  28  cents.  We  take  the  lib.  at  28  cents,  and  thus  have 
a  mixture  of  the  average  rate,  by  having  taken,  in  all,  lib.  at  20  cents, 
lib.  at  23  cents,  lib.  at  26  cents,  and  2lb.  at  28  cents.  We  prove  the 
correctness  of  the  result  by  dividing  the  value  of  the  whole  -mixture, 
or  S  1.25,  by  the  number  of  pounds  taken,  or  5,  which  gives  25  cents, 
or  the  given  mean  price  per  pound. 

Having  arranged  in  a  column 
the  prices  of  the  articles  with  the 
given  mean  price  on  the  left,  we 
Ans.  connect  together  the  terms  denot- 
ing the  price  of  each  article,  so  that 
a  price  less  than  the  given  mean 
is  united  with  one  that  is  greater. 
We  then   proceed  to  find  what  quantity  of  each  of  the  two  kinds, 


SECOND   OPERATION. 


25cts.  < 


20cts 
23cts. 

2  Gets, 
28cts, 


] 


31b. 
lib. 
21b. 
51b. 


319.  What  is  allirration  alternate  ?    Explain  the  first  operation, 
proved  to  be  correct  ■?     How  do  you  connect  the  prices  1 


How  is  it 


302 


ALLIGATION  ALTEKNATE. 


whose  prices  have  been  connected,  can  be  taken,  in  making  a  mixture, 
so  that  what  shall  be  gained  on  the  one  kind  shall  be  balanced  by  the 
loss  on  the  other.  By  taking  lib.  of  spice  at  20  cents,  the  gain  will 
be  5  cents ;  and  by  taking  lib.  at  28  cents,  the  loss  will  be  3  cents. 
To  equalize  the  gain  and  loss  in  this  cjise,  it  is  evident  we  should  take 
as  many  more  pounds  of  that  at  28  cents  as  the  loss  on  lib.  of  it  is  less 
than  the  gain  on  lib.  of  that  at  20  cents ;  or,  in  other  words,  the 
quantity  of  the  articles  taken  should  he  in  the  inverse  ratio  (Art.  236) 
of  the  difference  between  their  respective  prices  and  the  given  mean  price. 
Therefore,  we  take  5lbs.  at  28  cents,  and  3lbs.  of  that  at  20  cents, 
and  the  loss,  3cts.  X  5  =  15  cents,  on  the  former,  exactly  oifsets  the 
gain,  5cts.  X  3  =  15  cents,  on  the  latter.  We  write  the  3lb.  against 
its  price,  20  cents ;  and  the  5lb.  against  its  price,  28  cents.  In  like 
manner  we  determine  the  quantity  that  may  be  taken  of  the  other  two 
articles,  whose  prices  are  connected,  by  finding  the  difference  between 
each  price  and  the  mean  price ;  and,  as  before,  write  the  quantity 
taken  against  its  price. 

We  obtain,  as  a  result,  3lb.  at  20  cents,  lib.  at  23  cents,  2lb.  at  26 
cents,  5lb.  at  28  cents ;  this,  in  the  same  manner  as  the  other  answer, 
may  be  proved  to  satisfy  the  conditions  of  the  cjuestion,  since  examples 
of  this  kind  admit  of  several  answers. 

Rule.  —  Write  the  prices  of  the  articles  in  a  column,  with  the  mean 
rate  on  the  left,  and  connect  the  rate  of  each  article  which  is  less  than  the 
given  mean  rate  ivith  one  that  is  greate^^ 

Write  the  difference  between  the  mean  rate  and  that  of  each  of  the 
articles  opposite  to  the  rate  with  tvhich  it  is  connected ;  and  the  number  set 
against  each  rate  is  the  quantity  of  the  article  to  be  taken  at  that  rate. 

Note.  —  There  will  be  as  many  different  answers  as  there  arc  different 
ways  of  connecting  the  prices,  and  by  multiplying  and  dividing  these  answers 
they  may  be  varied  indefiuitcly. 

Examples  for  Practice. 

2.  A  farmer  wishes  to  mix  corn  at  75  cents  a  bushel,  with  ryo 
at  60  cents  a  bushel,  and  oats  at  40  cents  a  bushel,  and  wheat  at 
95  cents  a  bushel ;  what  quantity  of  each  must  he  take  to  make  a 
mixture  worth  70  cents  a  bushel? 


HIRST  OPERATION.   SECOND  OPERATION. 

Ans 

40—  25 


THIRD   OPERATION. 


70 


GOt 
'  75J 
[95— 


70 


Ans. 

40-       5 

60- 

-  25 

75- 

30 

95- 



10 

y7o^ 


40-:- 

60n;  : 
75-).:  ; 


4-25  =  30 


5  =  30 


25 
5 

10 
95^-  304-10  =  40 


■25 

■30 


:40 


y  3 


320.  —  The  rule  for  allipation  alternate?     How  can  you  obtain  different 
answers'?     Arc  tiicy  all  true  ? 


ALLIGATION  ALTERNATE.  303 

3.  I  have  4  kinds  of  salt,  worth  25,  30,  40,  and  50  cents  per 
bushel ;  how  much  of  each  kind  must  be  taken,  that  a  mixture 
might  be  sold  at  42  cents  per  bushel? 

Ans.  8  bushels  at  25,  30,  and  40  cents,  and  31  bushels  at  50 
cents. 

321  •  When  the  quantity  of  one  article  i^  given  to  find 
the  quantity  of  each  of  the  others. 

Ex.  1.  How  much  sugar,  that  is  worth  6,  10,  and  13  cents  a 
pound,  must  be  mixed  with  20Ib.  worth  15  cents  a  pound,  so  that 
the  mixture  will  be  w^h  1 1  cents  a  pound  ? 


OPEKATION. 


"  \  J.& 


13- 

[15- 


41  i 

2  !    I  Then,  5  :  1 

Ml  5:2 

5  J  «  5:4 


20 
20 
20 


8  Y  Ans. 
16) 


By  the  conditions  of  the  question  we  are  to  take  201b.  at  15  cents  a 
pound  ;  but  by  the  operation  we  find  the  difference  at  15  cents  a  pound 
to  be  only  5lb.,  which  is  but  \  of  the  given  quantity.  Therefore,  if 
we  increase  the  5lb.  to  20,  the  other  differences  must  be  increased  in 
the  same  ratio.     Hence  the 

Rule.  —  Find  the  difference  between  the  rate  of  each  and  the  mean 
rate ;  then  say,  As  the  difference  of  that  article  whose  quantity  is  given  is 
to  each  of  the  differences  separately,  so  is  the  quantity  given  to  the  several 
quantities  required. 

Examples   for   Practice. 

2.  A  farmer  has  oats  at  50  cents  per  bushel,  peas  at  GO  cents, 
and  beans  at  $  1.50.  These  he  wishes  to  mix  with  30  bushels  of 
corn  at  $  1.70  per  bushel,  that  he  may  sell  the  whole  at  %  1.25 
per  bushel ;  how  much  of  each  kind  must  he  take  ? 

Ans.  18  bushels  of  oats,  10  bushels  of  peas,  and  26  bushels  of 
beans. 

3.  A  merchant  has  two  kinds  of  sugar,  one  of  which  cost  him 
10  cents  per  lb.,  and  the  other  12  cents  per  lb.;  he  has  also 
1001b.  of  an  excellent  quality,  which  cost  him  15  cents  per  lb. 
Now,  as  he  ought  to  make  25  per  cent,  on  liis  cost,  how  much  of 
each  quantity  must  be  taken  that  he  may  sell  the  mixture  at  14 
cents  per  lb.  ? 

Ans.  383^1b.  at  10  cents,  and  100  lb.  at  12  cents. 

321.  What  is  the  rule  for  finding  the  quantity  of  each  of  the  other  articles 
■fflien  one  is  given'?  % 


304 


ALLIGATION  ALTERNATE. 


322.  "When  the  sum  of  the  quantities  of  the  articles 
and  their  mean  rate  are  given,  to  find  what  quantity  of 
eacli  must  be  taken. 

Ex.  1.  I  have  teas  at  25  cents,  35  cents,  50  cents,  and  70  csnts 
a  pound,  with  which  I  wish  to  make  a  mixture  of  1801b.,  that  will 
be  worth  45  cents  a  pound.  How  much  of  each  kind  must  I 
take  ? 

OPERATION. 


45 


25-^ 
35 
50 
70 


J 


25     Then,  60 


5 

10 
20 


60 
60 
60 


25 
5 

10 
20 


180 
180 
180 
180 


75 
15 
30 
60 


Ans. 


Sum  of  diffei-ences,  60 


Proof,         180 


By  the  conditions  of  the  question,  the  weight  of  the  mixture  is 
1801b.,  but  by  the  operation  we  find  the  sum  of  the  differences  to  be 
only  60lb.,  which  is  but  i-  of  the  quantity  required.  Therefore,  if  we 
increase  60lb.  to  180,  each  of  the  differences  must  be  increased  in  the 
same  ratio,  in  order  to  make  a  mixture  of  180lb.,  the  quantity  re- 
quired.    Hence  the 

Rule.  —  Find  ilie  differences  as  before ;  then  say,  As  the  suin  of  the 
differences  is  to  each  of  the  differences  separately,  so  is  the  given  quantity 
to  the  required  quantity  of  each  article. 

Examples  for  Practice. 

2.  John  Smith's  "  great  box  "  will  hold  100  bushels.  He  has 
wheat  worth  $2.50  per  bushel,  and  rye  worth  $2.00  per  bushel. 
How  much  chaff,  of  no  value,  must  he  mix  with  the  wheat  and 
rye,  that,  if  he  till  the  box,  a  bushel  of  the  mixture  may  be  sold 
at  $  1.80  ? 

Ans.  40bu.  each  of  wheat  and  rye,  and  20bu.  of  chaff. 

3.  I  have  two  kinds  of  molasses,  which  cost  me  20  and  30  cents 
per  gallon  ;  I  wish  to  fill  a  hogshead,  that  will  hold  80  gallons, 
willUhese  two  kinds.  How  much  of  each  kind  must  be  taken, 
that  I  may  sell  a  gallon  of  the  mixture  at  25  cents  per  gallon, 
and  make  10  per  cent,  on  my  purchase? 

Ans.  58t\  of  20  cents,  and  21^9x  of  30  cents. 

4.  I  have  sugars  at  10  cents  and  15  cents  ]ier  pound.  How 
much  of  each  must  be  taken,  that  a  mixture  containing  60  pounds 
shall  be  worth  $  7.20  ? 

Ans.  36  pounds  at  10  cents,  and  24  pounds  at  15  cents. 

322.  How  do  you  find  what  quantity  of  each  ingredient  must  be  taken 
when  the  sum  and  mean  price  are  yiven  ? 


PERMUTATION.  305 


PERMUTATION. 

323t  Pfrmutation  is  the  pi'ocess  of  finding  the  different  ordei'S 
in  which  a  given  number  of  things  may  be  ai'ranged. 

324,  To  find  the  number  of  different  arrangements 
that  can  be  made  of  any  given  number  of  things. 

Ex.  1.  How  many  different  numbers  may  be  formed  from  the 
figures  of  the  following  number,  432,  making  use  of  three  figures 
in  each  number  ?  Ans.  6. 

FIRST  OPERATION,  In  the  Ist  Operation,  we 

4  3  2,  4  2  3,  3  4  2,  3  2  4,  2  4  3,  2  3  4.    have  made  all  the  different 

arrangements  that  can  be 
SECOND  OPERATION.  made  of  the  given  figures, 

1X2X3  =  6.  ^^^  fi'^d  the  number  to  be 

6.  In  the  second  opera- 
tion, the  same  result  is  obtained  by  simply  multiplying  together  tbe 
first  three  of  the  digits,  a  number  equal  to  the  number  of  figures  to 
be  arranged.     Hence  the 

Rule.  —  Multiply  together  all  the  terms  of  the  natural  series  ofnum- 
bers,  from  1  up  to  the  given  number,  and  the  last  product  will  be  the 
answer  required. 

Examples  for  Practice. 

2.  My  family  consists  of  nine  persons,  and  each  person  has 
his  particulai"  seat  around  my  table.  Now,  if  their  situations 
were  to  be  changed  once  each  day,  for  how  many  days  could 
they  be  seated  in  a  different  position  ? 

Ans.  362880  days,  or  994  years  70  days. 

3.  On  a  certain  shelf  in  my  library  there  are  12  books.  If  a 
person  should  remove  them  without  noticing  their  order,  what 
would  be  the  probability  of  his  replacing  them  in  the  same  posi- 
tion they  were  at  first?  Ans.  1  to  479001600. 

4.  How  many  words  can  be  made  from  the  letters  in  the  word 
"  Embargo,"  provided  that  any  arrangement  of  them  may  be 
used,  and  that  all  the  letters  shall  be  taken  each  time  ? 

Ans.  5040  words. 

323.  What  is  permutation  1  — 324.  What  is  tlie  rule  for  finding  the  num- 
ber of  arrangements  that  can  be  made  of  any  given  number  of  things  1 
26* 


306 


MENSURATION   OF   SURFACES. 


MENSURATION    OF    SURFACES. 

325.  A  Surface  is  that  which  has  length  and  breadth  without 
thickness. 

The  Area  of  a  figure  is  its  surface  or  superficial  contents. 
A  Line  is  length  without  breadth  or- thickness. 
A  Plane  is  that  in  which,  if  any  two  points  be   taken,  the 
straight  line  that  joins  them  will  he  wholly  in  it. 

326.  An  Angle  is  the  inchnation  or  opening  of  two  lines,  which 
meet  in  a  point. 

The  Vertex  of  an  angle  is  the  pomt  of  meeting  of  the  hnes 
forming  the  anojle. 

A  Riglit  Angle  is  an  angle  formed  by  one  line 
falling  perpendicularly  on  another,  and  it  con- 
tains 90  degrees ;  as  A  B  C. 

*An  Acute  Angle  is  an  angle  less  than  a  right 
angle,  or  less  than  90  degrees ;  as  E  B  C 

An  Obtuse  Angle  is  an  angle  greater  than  a  right 
angle,  or  more  than  90  degrees ;  as  F  B  C. 

TRIANGLES. 

327.  A  Triangle  is  a  plane  figure  having  three  sides  and  three 
angles. 

It  receives  the  particular  names  of  an  equilateral  triangle,  isos- 
celes triangle,  and  scalene  triangle  ;  also,  of  a  right-angled  triangle, 
acute-angled  triangle,  and  obtuse-angled  triangle. 

The  Base  of  a  triangle,  or  other  plane  figure,  is  one  of  its  sides, 
on  which  it  may  be  supposed  to  stand  ;  as  C  D. 

The  Altitude  of  a  triangle  is  a  line  drawn  from  one  of  its  angles 
perpendicular  to  its  opposite  side  or  base  ;  as  A  B.  a 

An  Equilateral  Triangle  is  one  which  has  its  three 

sides  equal. 


.325.  Wh.at  is  a  surface  1  What  are  the  suporficial  contents  of  a  fijjurc 
called  7  —  326.  What  is  an  anj^lc  1  A  right  angle  ?  An  artite  angle  ?  An 
obtuse  angle  7 — 327.  What  is  a  triangle  ?  AVhat  particular  names  dues  it 
receive  1  When  is  it  called  a  right-angled  triangle  ?  An  acute-angled  tri- 
angle ?  An  obtuse-angled  triangle  7  What  is  the  base  of  a  triangle  7  The 
ttltitudo  7     What  13  an  equilateral  triangle  7 


MENSURATION  OF  SURFACES.  307 


An  Isosceles  Triangle  is  one  wliich  has  two  of  its 
sides  equal. 

A  Scalene  Triangle  is  one  which,  has  its  three  sides 
unequal. 

A  Right-Angled  Triangle  is  one  which  has  a  right 
angle 


o 


Note.  —  An  acute-angled  triangle  is  one  -which  has  an  acute  angle,  and  an 
obtuse-angled  triangle  is  one  having  an  obtuse  angle. 

328.    To  find  the  area  of  a  triangle. 

Rule  1.  —  Multiply  the  base  by  half  the  altitude,  and  the  product 
will  be  the  area.     Or, 

Rule  2.  —  Add  the  three  sides  together,  take  half  that  sum,  and  from 
this  subtract  each  side  separately ;  then  midtiply  the  half  of  the  sum  and 
these  remainders  together,  and  the  square  root  of  this  product  will  be 
the  area. 

1.  "What  are  the  contents  of  a  ti'iangle,  whose  base  is  24  feet, 
and  whose  jjerpendicular  hight  is  18  feet?  Ans.  216  feet. 

2.  What  are  the  contents  of  a  triangular  piece  of  land,  whose 
sides  are  50  rods,  60  rods,  and  70  rods  ? 

Ans.  1469.69+  rods. 

QUADRILATERALS. 

329t  A  Quadrilateral  is  a  plane  figure  having  four  sides,  and 
consequently  four  angles. 

It  comprehends  the  rectangle,  square,  rhombus,  rhomboid,  trape- 
zium, and  trapezoid. 

33®.  A  Parallelogram  is  any  quadrilateral  whose  opposite-  sides 
are  parallel. 

It  takes  the  particular  names  of  rectangle,  square,  rhombus,  and 
rhomboid. 

The  Allitllde  of  a  parallelogram  is  a  perpendicular  line  drawn 
between  any  two  of  its  opposite  sides  ;  as  C  D  in  the  rhomboid. 

327.  What  is  an  isosceles  tiianfrle  ?  A  scalene  triangle  1.  A  right-angled 
triangle'? — 328.  The  first  rule  for  finding  the  area  of  a  triangle?  The 
second'? — 329.  What  is  a  quadrilateral?  Wliat  figures  does  it  compre- 
hend ?  —  330  What  is  a  parallelogram  ?  Wliat  particular  names  does  it  take  'i 
The  altitude  of  a  parallelogram  ? 


308 


MENSURATION   OF   SURFACES. 


A  Rectangle  is  any  right-angled  parallelogram. 


A  Square  is  a  parallelogram,  having  equal  sides 
and  right  angles. 


A  Rliombuid  is  an  oblique-angled  parallelogram. 


A  Rhombus  is  an  oblique-angled  parallelogram, 
having  all  its  sides  equal. 


Note.  —  An  oblique  angle  is  one  either  acute  or  obtuse. 

331.    To  find  the  area  of  a  parallelogram. 

Rule.  —  Multiply  the  base  hi/  the  altitude,  and  the  product  loill  he 
the  area. 

1 .  What  are  the  contents  of  a  board  25  feet  long  and  3  feet 
wide  ?  Ans.  75  square  feet. 

2.  What  is  the  difference  between  the  contents  of  two  floors  ; 
the  one  being  37  feet  long  and  27  feet  wide,  and  the  other  40 
feet  long  and  20  feet  wide  ?  Ans.  199  square  feet. 

3.  Tlie  base  of  a  rhombus  is  15  feet,  and  its  perpendiculai 
hight  is  12  feet ;  what  are  its  contents  ? 

Ans.  180  square  feet. 


332.  A  Trapezoid  is  a  quadrilateral  which  has 
only  two  of  its  sides  parallel. 

333.  To  find  the  area  of  a  trapezoid. 

Rule.  —  Multiply  half  of  the  sum  of  the  parallel  sides  by  the  altitude, 
and  the  product  is  the  area. 

1.  What  is  the  area  of  a  trapezoid,  the  longer  parnllol  side 
being  482  feet,  the  shorter  324  feet,  and  the  .ilfitiide  21  fi  feet  ? 

Ans.  87018  square  feet. 


.•?3n.  What  is  a  reptnntrle  ?     A  sqnnro?     A  i-homboiil  ?     A  rhomhus  ? 

331.  The  ruin  for  findinfr  the  area  of  a  parMllcloirram  ? — .332.  What  is  a 
trapezoid  1  — 333.  What  is  the  rule  for  finding  the  area  of  a  trapezoid  f 


MENSURATION  OF  SURFACES.  309 

2.  What  is  the  area  of  a  plunk,  whose  length  is  22  feet,  the 
width  of  the  wider  end  being  28  inches,  and  of  the  narrower  20 
inches  ?  Ans.  44  square  feet. 

334.  A  Trapezium  is  a  quadrilateral,  which  has 
no  two  of  its  sides  parallel. 

A  Diagonal  is  a  straight  line  which  joins  the 
vertices  of  any  two  opposite  angles  of  a  quadi'i- 
lateral ;  as  E  F. 

335.  To  find  the  area  of  a  trapezium. 

Rule.  —  Divide  the  trapezium  into  two  triangles  hy  a  diagonal^  and 
then  find  the  areas  of  these  triangles  ;  their  sum  will  be  the  area  of  the 
trepezium. 

1.  "What  is  the  area  of  a  trapezium,  whose  diagonal  is  65  feet, 
and  the  lengths  of  the  perpendiculars  .let  fall  upon  it  are  14  and 
18  feet  ?  .  Ans.  1040  square  feet. 

2.  What  is  the  area  of  a  trapezium,  whose  diagonal  is  125 
rods,  and  the  lengths  of  the  perpendiculars  let  fall  upon  it  are  7(? 
and  85  rods  ?  Ans.  9687.5  square  rods. 

POLYGONS. 

336.  A  Polygon  is  any  figure  bounded  by  straight  lines. 

It  takes  the  particular  names  of  pentagon,  when  it  is  a  polygon 
of  five  sides  ;  hexagon,  one  of  six  sides  ;  heptagon,  one  of  seven 
sides ;  octagon,  one  of  eight  sides ;  nonagon,  one  of  nine  sides  ; 
decagon,  one  of  ten  sides ;  undecagon,  one  of  eleven  sides ;  and 
dodecagon,  one  of  twelve  sides. 

337.  A  Regular  Polygon  is  one  which  has  all  its 
sides  and  all  its  angles  equal. 

The  Perimeter  of  a  polygon  is   the  broken   line 
which  bounds  it. 

338.  To  find  the  area  of  a  regular  polygon. 

Rule.  —  Multiply  the  perimeter  hy  half  the  perpendicular  let  fall  from 
the  center  on  one  of  its  sides,  and  the  product  will  be  the  area. 

334.  What  is  a  trapezium  ?  What  is  a  diagonal  ?  —  33.5.  The  rule  for 
finding  the  area  of  a  trapezium  ?  —  336.  What  is  a  polygon  ?  What  par- 
ticular names  does  it  take  1  —  337.  What  is  a  regular  polygon  1 


310  MENSURATION  OF  SURFACES. 

1.  Wliat  is  the  area  of  a  regular  pentagon,  whose  sides  are 
each  35  feet,  and  the  perpendicular  24.08  feet  ? 

Ans.  2107  square  feet. 

2.  What  is  the  area  of  a  regular  hexagon,  whose  sides  are 
each  20  feet,  and  the  perpendicular  17.32  feet  ? 

Ans.  1039.20  square  feet. 

THE  CIRCLE. 

339.  A  Circle  is  a  plane  figure  bounded  by  a 
curved  line,  every  part  of  which  is  equally  distant 
from  a  point,  called  its  center.  ^  I 

The  Circumference  or  Periphery  of  a  circle  is  the 

line  which  bounds  it. 

The  Diameter  of  a  circle  is  a  line  drawn  through  the  center, 
and  tenninated  by  the  circumference  ;  as  G  H. 

340.  To  find  the  circumference  of  a  circle,  the  diam- 
eter being  given. 

KuLE.  —  MuUijjly  the  diameter  by  3.141592,  a/ic?  the  product  is  the 
circumference. 

Note.  —  3.141592  is  the  circumference  of  a  circle  whose  diameter  is  1. 
(Art.  291.) 

1.  What  is  the  circumference  of  a  circle,  whose  diameter  is 
50  feet  ?  Ans.  157.0796-j-  feet. 

2.  A  gentleman  has  a  circular  garden  whose  diameter  is  100 
rods ;  what  is  the  length  of  the  fence  necessary  to  enclose  it  ? 

Ans.  314.15-|-  rods. 

341.  To  find  the  diameter  of  a  circle,  the  circumfer- 
ence being  given. 

Rule.  —  Multiply  the  circumference  by  .318309,  and  the  product  will 
be  the  diameter. 

Note. — .318309  is  the  diameter  of  a  circle  whoso  circumference  is  1. 
(Art.  291.) 

1.  What  is  the  diameter  of  a  circle,  whose  circumference  is  80 
miles  ?  Ans.  25.46-j-  miles. 

2.  If  the  circumference  of  a  wheel  is  62.84  feet,  what  is  the 
diameter  ?  Ans.  20-|-  feet. 

.337.  What  is  the  perimeter  of  a  polygon  7  —  338.  Tlic  rule  for  findiiic:  tho 
area  of  a  roijnlar  pol yfcon  7  —  339.  What  is  a  circle  ?  Tlic  ciminifiTence  of  a 
circle  7  The  rliameter  of  a  circle?  —  340.  The  rule  for  fiiidinc  the  circum- 
feronce  of  a  circle,  the  diameter  heinp:  piven  7  —  34 1 .  The  rule  for  finding  tho 
diameter  of  a  circle,  the  circumference  being  given  ? 


MENSUKATION   OF   SURFACES.  311 

342.  To  find  the  area  of  a  circle,  the  diameter,  the 
drcumference,  or  both,  being  given. 

Rule  1.  —  Multiply  half  the  diameter  hj  half  the  circumference,  and 
the  product  is  the  area. 

EuLE  2.  —  Multiply  the  square  of  the  diameter  by  .785398,  and  the 

product  is  the  area. 

Note. .785398,  or  i  of  3.141592,  is  the  area  of  a  circle  whose  diameter 

is  1.     (Art.  291.) 

1.  If  the  diameter  of  a  circle  be  200  feet,  what  is  the  area  ? 

Ans.  31415.92  square  iieet. 

2.  There  is  a  certain  farm,  in  the  form  of  a  circle,  whose  cir- 
cumference is  400  rods ;  how  many  acres  does  it  contain  ? 

Ans.  79A.  2R.  12+p. 

343.  To  find  the  side  of  a  square  equal  in  area  to  a 
given  circle. 

The  square  in  the  figure  is  supposed  to  have  the 
same  area  as  the  circle. 

EuLE.  —  Multiply  the  diameter  hy  .886227,  and  the  product  is  the  side 
of  an  equal  square. 

Note.  —  .886227,  or  the  square  root  of  .785398,  is  the  side  of  a  square 
which  is  equivalent  to  a  circle  whose  diameter  is  1.     (Art.  292.) 

1.  We  have  a  round  field  40  rods  in  diameter ;  what  is  the 
side  of  a  square  field  that  will  contain  the  same  quantity  ? 

Ans.  35.44-[-  rods. 

2.  I  have  a  circular  field  100  rods  in  circumference ;  what 
must  be  the  side  of  a  square  field  that  shall  contain  the  same 
area  ?  Ans.  28.2-[-  rods. 

344.  To  find  the  side  of  a  square  inscribed  in  a  given 
circle. • 

A  square  is  said  to  be  inscribed  in  a  circle  when 
the  vertices  of  its  angles  are  in  the  circumference. 


342.  The  rule  for  finding  the  area  of  a  circle,  when  the  diameter  is  triven? 
"When  the  circumference  is  given  1  When  the  diameter  and  circumference 
are  both  given  ?  —  343.  The  rule  for  finding  the  side  of  a  square  equal  in  area 
to  a  given  circle  ?  —  344.  When  is  a  square  said  to  be  inscribed  in  a  circle  1 


312  MENSUExVTION   OF   SOLIDS. 

Rule.  —  Mulliply  the  diameter  hy  .707106,  and  tlie  product  is  the  side 
of  the  square  inscribed. 

Note.  — .707106  is  the  side  of  the  inscribed  square,  when  the  diameter 
of  the  circumscribed  circle  is  1.     (Ai-t.  292.) 

1.  What  is  the  thickness  of  a  square  stick  of  timber  that  may 
be  hewn  from  a  log  30  inches  in  diameter  ? 

Ans.  21.21-f-  inches. 

2.  How  large  a  square  field  may  be  inscribed  in  a  circle  whose 
circumference  is  100  rods?  Ans.  22.54-  rods  square. 

THE  ELLIPSE. 

345.  An  Ellipse  is  an  oval  figure  having  two 
diameters,  or  axes,  the  longer  of  which  is  called  the 
transverse  and  the  shorter  the  conjugate  diameter. 

346.  To  find  the  area  of  an  ellipse. 

Rule.  —  Multiply  the  two  diameters  together,  and  their  product  hy 
.785398  ;  the  last  product  is  the  area. 

1.  What  is  the  area  of  an  ellipse  whose  transverse  diameter 
is  14  inches,  and  its  conjugate  diameter  10  inches  ? 

Ans.  109.95-]-  square  inches. 

2.  What  is  the  area  of  an  elliptical  table,  8  feet  long  and  5 
feet  wide?  Ans.  31  square  feet,  59-|-  square  inches. 


MENSURATION    OF    SOLIDS. 

347.  A  Solid,  or  Body,  is  that  which  has  length,  breadth,  and 
thickness. 

Mensuration  of  solids  includes  two  operations  :  first,  to  find 
their  superficial  contents,  and  second,  their  solidity  or  volume. 

THE  PRISM. 

348.  A  Prism  is  a  solid  whose  ends  are  any  plane  figures 
which  arc  equal,  similar,  and  parallel  to  each  other,  and  whose 
bides  ai'e  parallelograms. 


344.  The  rule  for  finding  the  side  of  a  square  inscribed  in  a  circle  ?  —  345. 
What  is  an  ellipse  1  What  is  the  longer  diameter  called  ?  The  shorter?  — 
."^46.  The  rule  for  finding  the  area  of  an  ellipse  •? —  347.  What  is  a  solid  ? 
What  two  operations  does  meusiaation  of  solids  include  ?  —  348.  What  is  a 


prism  1 


MENSURATION   OF   SOLIDS. 


818 


It  takes  particular  names,  according  to  tlie  figure  of  its  base  or 
ends,  namely,  triangular  prism,  square  prism,  pentagonal  prism, 
&c. 

The  Base  of  a  prism  is  either  end ;  and  of  solids  in  general,  the 
part  upon  which  they  are  supposed  to  stand. 

All  prisms  whose  bases  are  parallelograms  are  comprehended 
under  the  general  name  Paralklopipcdoas  or  Parallclopipeds. 


A  Tfiansular  Prism  is  one  whose  base  is  a  triangle. 


A  Square  Prism  is  one  whose  base  is  a  square,  and 
when  all  the  sides  are  squares  it  is  called  a  cube. 


A  Pentagonal  Prism  is  one  whose  base  is  a  pentagon. 


349.   To  find  the  surface  of  a  prism. 

Rule.  —  Multiply  the  perimeter  of  its  base  by  its  MgTit,  and  to  this  pro- 
duct add  the  area  of  the  two  ends ;  the  sum  is  the  area  of  the  prism. 

1.  "What  are  the  superiicial  contents  of  a  triangular  prism, 
the  width  of  whose  side  is  3  feet,  and  its  length  15  feet? 

Ans.  142.79-j-  square  feet. 

2.  What  is  the  surface  of  a  square  prism,  whose  side  is  9  feet 
wide,  and  its  length  25  feet?  Ans.  1062  square  feet. 

35d»   To  find  tlie  solidity  of  a  prism. 

Rule.  —  Multiply  the  area  of  the  base  by  the  hight,  and  the  product  is 
the  solidity. 

348.  "What  particular  names  does  the  prism  tako.l  What  is  the  base  of  a 
prism  and  of  solids  in  general?  What  is  a  parallelopipcd  or  parallelopipe- 
don  ?  What  is  a  trianprular  prism  1  A  square  prism  ?  A  pentagonal 
prism  ?  —  349.  The  rule  for  finding  the  surface  of  a  prism  ■?  — 350.  The  rule 
for  finding  the  soliditj'  of  a  prism  1 

27 


314  MENSURATION    OF   SOLIDS. 

1.  What  are  the  contents  of  a  triangular  prism,  whose  length 
is  20  feet,  and  the  three  sides  of  its  triangular  end  or  base  5,  4, 
and  3  feet?  •  Ans.  120  cubic  feet. 

2.  How  many  cubic  feet  are  there  in  a  cube,  whose  sides  are 
8  feet?        ■  Ans.  512  cubic  feet. 

3.  What  is  the  number  of  cubic  feet  in  a  room  30  feet  long,  20 
feet  wide,  and  10  feet  high  ?  Ans.  6000  cubic  feet. 

THE   CYLINDER. 

3.51.    A  Cylinder  is  a  round  body,  of  uniform  diameter, 
with  circular  ends  or  bases  parallel  to  each  other. 

The  Axis  of  a  cylinder  is  a  straight  line  drawn  through 
it,  from  the  center  of  one  end  to  the  center  of  the  other. 

352.  To  find  the  surface  of  a  cylinder. 

Rule.  —  Multiply  the  circumference  of  the  base  hj  the  altitude,  and  to 
the  product  add  the  areas  of  the  two  ends ;  the  sum  icill  he  the  whole  sur- 
face. 

1.  What  is  the  surface  of  a  cylinder,  whose  length  is  4  feet,  and 
the  circumference  3  feet?  Ans.  13.43 -]-  square  feet. 

2.  John  Snow  has  a  roller  12  feet  long  and  2  feet  in  diameter ; 
what  is  its  convex  surface?  Ans.  75.39-j-  square  feet. 

353.  To  find  tlie  solidity,  or  volume,  of  a  cylinder. 

Rule.  —  Multiply  the  area  of  the  base  by  the  altitude,  and  the  product 
will  be  the  solidity  or  volume. 

1.  What  is  the  solidity  of  a  cylinder  8  feet  in  length  and  2 
feet  in  diameter  ?  Ans.  25.13-)-  cubic  feet. 

•     2.  What  is  the  solidity  of  a  cylinder,  who^e  diameter  is  5  \eet, 
and  its  altitude  20  feet?  Ans.  392.G9-J-  cubic  feet. 

THE  PYRAJIID  AND   CONE. 

354.  A  Pyramid  is  a  solid,  standing  on  a  triangular, 
square,  or  polygonal  base,  with  its  sides  tapering  uni- 
forndy  to  a  point  at  the  top,  called  the  vertex. 

The  Slant  Iligllt  of  a  pyramid  is  a  line  drawn  from        ; 
the  vertex  to  the  middle  of  one  of  the  sides  of  the  ;\ 

bi^se.  v_n_i/ 

3:')!.  Wluit  is  a  rylindcr?  Wliat  i.^  the  axis  of  a  cvliiulcr  ?  —  3f>2.  Tho 
rule  for  (iiKJinjr  the"  siufiU'c  of  n  cvlinder?  —  :iv>^.  The  rule  for  (iiiirm,<,'  the 
soliiliiy  of  a  cylinder  ?  —35-i.  What  is  a  pynunid  ?  Tho  slant  hight  of  a 
pyrainid  i 


MENSURATION   OF   SOLIDS.  815 

355 •  A  Cone  is  a  solid,  having  a  circle  for  its  base, 
and  tapering  uniformly  to  a  point,  called  the  vertex. 

The  Altitude  of  a  pp-amid  and  of  a  cone  is  a  line 
drawn  from  the  vertex  perpendicular  to  the  plane  of  the 
base  ;  as  B  C. 

The  Slant  night  of  a  cone  is  a  line  drawn  from  the  ^ 
vertex  to  the  circumference  of  the  base ;  as  A  C. 

356.  To  find  the  convex  surface  of  a  pyramid  or  of  a 
colie. 

Rule.  —  Multiply  Ihe  perimeter  or  the  circumference  of  the  base  by 
half  its  slant  hiyht,  and  the  product  is  the  convex  surface. 

1.  How  many  yards  of  cloth,  that  is  27  inches  wide,  will  it 
require  to  cover  the  sides  of  a  pyramid  whose  slant  hight  is  100 
feet,  and  whose  perimeter  at  the  base  is  54  feet  ? 

Ans.  400  yards. 

2.  Required  the  convex  surface  of  a  cone,  whose  slant  hight 
is  50  feet,  and  the  circumference  at  its  base  12  feet. 

Ans.  300  square  feet. 

357.  To  find  tlie  solidity  or  volume  of  a  pyramid  or  of 
a  cone. 

Rule.  —  Multiply  the  area  of  the  base  by  one  third  of  its  altitude,  and 
the  product  will  be  the  solidity. 

1.  The  largest  of  the  Egyptiafi  pyramids  is  square  at  its  base, 
and  measures  693  feet  on  a  side.  Its  altitude  is  500  feet.  Now, 
supposing  it  to  come  to  a  point  at  its  vertex,  what  are  its  solid 
contents,  and  how  many  miles  in  length  of  wall  Avould  it  make, 
4  feet  in  hight  and  2  feet  thick  ? 

Ans.  80041500  cubic  feet ;  1894.9  miles  in  length. 

2.  Wliat  are  the  solid  contents  of  a  cone,  whose  hight  is  30 
feet,  and  the  diameter  of  its  base  5  feet  ?       Ans.  196.3-j-  feet. 

358.  A  Frnstnm  of  a  Pyramid  is  the  part  that  re- 
mains after  cutting  off  the  top,  by  a  plane  parallel 
to  the  base. 


355.  What  is  a  cone  ?  "What  is  the  altitude  of  a  pyramid  and  of  a  cone  ? 
The  slant  hi^rht  of  a  cone?  — .356.  The  rule  for  finding  the  surface  of  a  pyr- 
amid and  of  a  cone  1  —  357.  The  rule  for  finding  the  solidity  of  a  pyramid 
and  of  a  cone  ?  —  358.  What  is  the  frustum  of  a  pyramid  1 


816  MENSURATION  OF   SOLIDS. 

359.  A  Fi'UStum  of  a  Cone  is  the  part  that  remains       M  |« 

after  cuttinnr  off  the  top,  by  a  plane  parallel  to  the  base,      im  -'*1 

"^      *•  ^  •  #y"i..l.ii'!i"!ii 

360.  To  find  the  surface  of  a  frustum  of  a  pyramid  or 
of  a  coue. 

Rule.  —  Add  the  perimeters  or  the  circumferences  of  the  two  ends  to- 
gether, and  multiply  this  sum  by  half  the  slant  higlit.  Then  add  the  areas 
of  the  two  ends  to  this  product,  and  their  sum  will  be  the  surface. 

1.  There  is  a  square  pyramid,  whose  top  is  broken  off  20  feet 
slant  hight  from  the  base.  The  length  of  each  side  at  the  base 
is  8  feet,  and  at  the  top  4  feet ;  what  is  its  whole  surface  ? 

Ans.  560  square  feet. 

2.  There  is  a  frustum  of  a  cone,  whose  slant  hight  is  12  feet, 
the  circumference  of  the  base  18  feet,  and  that  of  the  upper  end 
9  feet ;  what  is  its  whole  surface  ? 

Ans.  194.22-}-  square  feet. 

361.  To  find  the  solidity  or  volume  of  a  frustum  of  a 
pyramid  or  of  a  cone. 

Rule.  —  Find  the  area  of  the  two  ends  of  the  frustum ;  multiply  these 
two  areas  together,  and  extract  the  square  root  of  the  product.  To  this 
root  add  the  two  areas,  and  multiply  their  sum  by  one  third  of  the  altitude 
of  the  frustum  ;  the  product  will  be  the  solidity. 

1.  What  is  the  solidity  of  ^he  frustum  of  a  square  pyramid, 
whose  hight  is  30  feet,  and  whose  side  at  the  bottom  is  20  feet, 
and  at  the  top  10  feet?  Ans.  7000  cubic  feet. 

2.  What  are  the  contents  of  a  stick  of  timber  20  feet  long,  and 
the  diameter  at  the  large  end  being  12  inches,  and  at  the  smaller 
end  6  inches  ?  Ans.  9.162-j-  feet. 

THE   SPHERE. 

362.  A  Sphere  is  a  solid,  bounded  by  one  continued 
convex  surface,  every  part  of  which  is  equally  distant 
from  a  point  within,  called  the  center. 

The  Axis  or  Diameter  of  a  sphere  is  a  line  passing 
through  the  center,  and  terminated  by  the  surface. 

359.  What  is  the  frustum  of  a  cone?  —  360.  The  rule  for  findinii  the  sur- 
face of  a  frustum  of  a  pyramid  or  of  a  coue?  —  361.  The  rule  for  finding 
the  solidity  of  a  frustum  of  a  ]>vriimid  or  of  a  coue  1  —  362.  What  is  a 
ephcre  1     The  diameter  or  axis  ol  a  sphera  ? 


MENSURATION   OF   SOLIDS.  317 

363i    To  find  the  surface  of  a  sphere. 

KuLE.  — Muhiphj  the  diameter  hy  the  circumference,  and  the  product 
will  be  the  surface. 

1.  What  is  the  convex  surface  of  a  globe,  whose  diameter  is 
20  inches  ?  Ans.  1256.G-}-  square  inches. 

2.  "if  the  diameter  of  the  earth  is  8000  miles,  wliat  is  its  con- 
vex surface  ?  Ans.  2010G1888  square  miles. 

364,    To  find  the  solidity  or  volume  of  a  sphere. 

Rule.  —  Multiply  the  surface  hj  \  of  the  diameter,  or  multiply  the 
cube  of  the  diameter  by  .523598,  and  the  product  will  be  the  solidity. 

Note.  — .523598  is  \  of  3.141592. 

1.  "What  is  the  solidity  of  a  sphere  whose  diameter  is  20 
inches  ?  Ans.  4188.7-j-  inches. 

2.  If  the  diameter  of  a  globe  or  sphere  is  5  feet,  how  many 
cubic  feet  does  it  contain  ?  Ans.  65.44-}-  cubic  feet. 

3S5.  To  find  how  large  a  cube  may  be  cut  from  any 
given  sphere,  or  be  inscribed,  in  it. 

Rule.  —  Square  the  diameter  of  the  sphere,  divide  the  product  by  3, 
and  extract  the  square  root  of  the  quotient  for  the  answer. 

1.  How  large  a  cube  may  be  inscribed  in  a  sphere  10  inches 
in  diameter  ?  Ans.  5.773-}-  inches. 

2.  AVliat  is  the  side  of  a  cube  that  may  be  cut  from  a  sphere 
30  inches  in  diameter  ?  Ans.  17.32-}-  feet. 

THE   SPHEROID. 

388.    A  Spheroid  is   a   solid,   generated  by  the       ^ss^^^„^ 
revolution  of  an  ellipse  about  one  of  its  diameters.        ^g^.^^m 

If  tlie  ellipse  revolves  about  its  longer  or  trans-      ^^fe;^^^ 
verse  diameter,  the  spheroid  is  prolate,  or  phlong  ; 
if  about  its  shorter  or  conjugate  diameter,  the  spheroid  is  oblate,  or 
Jiattened. 

367.    To  find  the  solidity  or  volume  of  a  spheroid. 

Rule.  —  Multiply  the  square  of  the  shorter  axis  by  the  longer  axis,  and 
tJds  product  by  .523598,  if  the  spheroid  is  prolate.     Or, 

363.  The  rule  for  finding  the  surface  of  a  sphere?  —  364.  The  rule  for  find- 
ing the  solidity  of  a  sphere  ?  —  365.  The  rule  for  finding  Itow  large  a  cube 
can  be  cut  from  a  given  sphere  ?  —  366.  What  is  a  spheroid  ?  A  prolate 
spheroid  ?  An  oblate  spheroid  1  —  367.  The  rule  for  finding  the  solidity  of 
a  spheroid  1 

27* 


318  MENSURATION   OF    LUMBER. 

Jf  it  is  oblate,  multiply  the  square  of  the  longer  axis  by  the  shorter  axis, 
and  this  product  by  .523598. 

1.  What  is  the  solidity  of  a  prolate  spheroid,  whose  ti'ansverse 
axis  is  30  feet,  and  the  conjugate  axis  20  feet  ? 

Ans.  6283.17-1-  cubic  feet. 

2.  What  is  the  solidity  of  an  oblate  spheroid,  whose  axes  are 
30  and  10  feet  ?  Ans.  4:712.38+  cubic  feet. 


MENSURATION    OF    LUMBER. 

368.  Boards  are  usually  measured  by  the  square  foot.  Planks, 
joists,  beams,  &c ,  are  usually  surveyed  by  board  measm-e,  the 
board  being  considered  to  be  1  inch  in  thickness. 

Round  timber  is  sometimes  measured  by  the  ton,  and  some- 
times by  boai'd  measure. 

369.  To  find  the  contents  of  a  board. 

Rule.  —  Multiply  the  length  of  the  hoard,  taken  in  feet,  by  its  icidth, 

taken  in  inches,  and  divide  this  product  by  12;  the  quotient  is  the  contents 

in  square  feet. 

Note.  —  If  the  board  is  tapering,  take  half  the  sum  of  the  width  of  its 
ends  for  the  width. 

1.  What  are  the  contents  of  a  board  18  inches  wide  and  16 
feet  lona:  ?  Ans.  2-4  feet. 

o 

2.  What  are  the  contents  of  a  board  2-4  feet  long  and  30  inches 
wide  ?  Ans.  60  feet. 

370.  To  find  the  contents  of  joists,  beams,  etc. 

Rule.  —  Multiply  the  loidth,  taken  in  inches,  by  the  thickness,  and  this 
product  by  the  length,  infect ;  divide  the  last  product  by  12,  and  the  quo- 
tient is  the  contents  in  feet. 

1.  What  are  the  contents  of  a  joist  4  inches  wide,  3  inches 
thick,  and  12  feet  long?  Ans.  12  feet. 

2.  What  are  the  contents  of  a  square  stick  of  timber  25  feet 
long  and  10  inches  thick?  Ans.  208j  feet. 

371.  To  find  the  contents  of  round  timber. 

Rule.  —  Multiply  the  length  of  the  stick,  taken  in  feet,  by  the  square 
of  one  fourth  the  girt,  taken  in  inches  ;  divide  thii  product  by  144,  and  the 
quotient  is  the  contents  in  cubic  feet. 

368.  By  what  measure  are  planks,  joists,  &c.,  nsuallv  surveyed  1  What  is 
the  usual  thickness  of  a  hoard?  How  is  round  timber  measured  7  —  369. 
The  rule  for  findinz  the  contents  of  a  board  ?  —  370  The  rule  for  tiudiug  tba 
contents  of  joist:*,  ic. 


MlSCELLAiSEOUS    QUESTIONS.  319 

Note  1.  —  Tlie  girt  of  tii|)cring  timber  is  usually  taken  about  one  tliird 
the  distance  from  tiie  larger  to  the  smaller  end. 

XoTE  2.  —  A  ton  of  timber,  estimated  by  this  method,  contains  50-1'^^ 
cubic  feet. 

1.  How  many  cubic  feet  of  timber  in  a  stick,  whose  length  is 
50  feet,  and  whose  girt  is  60  inches?  Ans.  78^  cubic  feet. 

2.  What  are  the  contents  of  a  stick,  whose  length  is  30  feet, 
and  girt  30  inches  ?  Ane.  11.7-|-  toiid  feet. 


MISCELLANEOUS    QUESTIONS. 

1.  What  number  is  that,  to  which  if  |  be  added,  the  sum  will 
be  7^?  Ans.  7  3-. 

2.  What  number  is  that,  from  which  if  of  be  taken,  the  re- 
mainder will  be  41  ?  '  Ans.  7^|-. 

3.  What  number  is  that,  to  which  if  3»  be  added,  and  the  f  um 
divided  by  5^,  the  quotient  will  be  5  ?  Ans.  23|. 

4.  From  -j-^^  of  a  mile  take  J  of  a  furlong. 

Ans.  4fur.  12rd.  8ft.  Sin. 

5.  John  Swift  can  travel  7  miles  in  f  of  an  hour,  but  Thomas 
Slow  can  travel  only  a  miles  in  -Jj  of  an  hour.  Both  started  from 
Danvei's  at  the  same  lime  for  Boston,  the  distance  being  12  miles. 
How  much  sooner  will  Swift  arrive  in  Boston  than  Slow  ? 

Ans.  12||-  seconds. 
G.  If  I  of  a  ton  cost  $49,  what  will  Icwt.  cost? 

An?.  $3.92. 

7.  How  many  bricks  8  inches  long,  4  inches  wide,  and  2 
inches  thick,  will  it  tukc  to  build  a  wiul  40  feet  long,  20  feet 
high,  and  2  feet  thick  ?  Ans.  43200  bricks. 

8.  How  many  bricks  will  it  take  to  build  the  walls  of  a  house, 
v.-hich  is  80  feet  long.  40  feet  wide,  and  25  feet  high,  the  wall  to 
be  12  inches  thick  ;  the  brick  being  of  the  same  dimensions  as  in 
(he  last  question  ?  Ans.  159300  bricks. 

9.  How  many  tiles,  8  inches  square,  Avill  cover  a  floor  18  feet 
long,  and  12  feet  wide?  Ans.  486  tiles. 

10.  If  it  cost  $  18.25  to  carry  llcwt.  3qr.  191b.  46  miles,  how 
much  must  be  paid  for  carrying  83cwt.  2qr.  111b.  96  miles? 

Ans.  $  266.70|5|a. 

11.  A  merchant  sold  a  piece  of  cloth  for  $  24,  and  thereby  lost 
25  per  cent. ;  what  would  he  have  gained  had  he  sold  it  for  $  34  ? 

Ans.  6:^  per  cent. 


820 


MISCELLANEOUS   QUESTIONS. 


12.  Bought  a  lio,2j5head  of  molasse?,  containing  120  gallons,  for 
$  30  ;  but  20  gallons  having  leaked  out,  for  what  must  I  sell  the 
remainder  per  gallon  to  gain  $10  ?  Ans.  $0.40. 

13.  Bought  a  quantity  of  goods  for  $  12R.25,  and  having  kept 
them  on  hand  6  mouths,  for  what  must  I  sell  them  to  gain  6  per 
cent.?  Ans.  $140.02. 

14.  If  a  sportsman  spends  ^  of  his  time  in  smoking,  ^  in  gun- 
ning, 2  hours  per  day  in  loafing,  and  G  houafs  in  eating,  drinking, 
and  sleeping,  how  much  remains  lor  useful  purposes  ? 

Ans.  2  hours. 

15.  If  a  lady  spend  ^  of  her  time  in  sleep,  ^  in  making  calls,,  ^ 
at  her  toilet,  |  in  reading  novels,  and  2  hours  each  day  in  receiv- 
ing visits,  how  large  a  portion  of  her  time  will  remain  for  improv- 
ing her  mind,  and  for  domestic  employments  ? 

Ans.  3 II  hours  per  day. 

16.  If  5|  ells  English  cost  $  15.1  G,  what  will  71  f  yards  cost? 

Ans.  ■$  155.39. 

17.  If  a  staff  4  feet  long  cast  a  shadow  5|  feet,  what  is  the 
hi"-ht  of  a  steeole  whose  shadow  is  150  feet?      Ans.  107f  feet. 

18.  Borrowed  of -James  Day  $  150  for  rix  month>  ;  afterwards 
I  lent  him  $  100  ;  how  long  shall  he  keep  it  to  comi)ensate  him 
for  the  use  of  the  sum  he  lent  me  ?  Ans.  9  mouths. 

19.  A  certain  town  is  taxed  $  G045.50 ;  the  valuation  of  the 
town  is  $  293275.00  ;  there  are  150  polls  in  the  town,  which  are 
taxed  $  1.20  each.  AVhat  is  the  tax  on  a  dollar,  and  what  does 
A  pay,  who  has  4  polls,  and  whose  property  is  valued  at  $3G75? 

Ans.  $  0.02.     A's  tax  $  78.30. 

20.  D.  Sanborn's  garden  is  234  rods  long,  and  134  rods  w!d(>, 
and  is  surrounded  by  a  good  fence  7|  feet  high.  Now,  if  he  shall 
make  a  walk  around  his  garden  within  the  fence  l-f'.j  feet  wide, 
how  much  will  remain  for  cultivation  ? 

Ans.  lA.  3R.  7p.  85{i|-].ft. 

21.  J.  Ladl's  garden  is  100  feet  long  and  80  feet  wide;  he 
wishes  to  enclose  it  with  a  ditch,  to  be  dug  outside,  4  feet  wide ; 
how  deep  must  it  be  dug,  that  the  soil  taken  from  it  and  i)laced 
Oil  the  garden  may  raise  the  surl'ace  1  foot?         Ans.  5jf  feet. 

22.  How  many  yards  of  paper,  that  is  30  inches  wide,  will  it 
require  to  cover  the  walls  of  u  room  that  is  15 J-  feet  long,  11^  feet 
wide,  and  7f-  feet  high?  Ans.  55^ J  yards. 

23.  Charles  Carleton  has  agreed  to  plaster  the  above  room, 
^valls  and  ccihng,  at  10  cents  per  square  yard  ;  w'hat  will  be  his 
bill  ?  Ans.  $  6.54i^. 


MISCELLANEOUS   QUESTIONS.  321 

24.  What  is  the  interest  of  $  17.SG,  from  Feb.  9,  1850,  to  Oct. 
29,  1852,  at  7^  per  cent.  ?  Ans.  $  3.52+. 

25.  Required  the  pnrface  of  the  largest  cube  that  can  be  in- 
scribed in  a  sphere  30  inches  in  diameter.      Ans.  1800  inches. 

26.  "What  is  due,  on  the  following  note,  at  compound  interest, 
Oct.  29,  18G2  ? 


$  1000.  Salem,  iV.  If.,  Oct.  29,  1856. 

F'or  value  received,  I  promise  to  pay  Luther  Emerson,  Jr.,  or 
order,  on  demand,  one  thousand  dollars  with  interest. 

...  .  Ejierson  Luther. 

Attest,  Adams  Ayer. 

On  this  note  are  the  following  indorsements  :  — 

Jan.  1,  1857,  was  received  $  125.00, 

June  5,  1857,         do.  ^316.00, 

Sept.  25,  1857,      do.  $417.00, 

April  1,  1858,        do.  $  100.00, 

July  7,  1858,         do.  %    50.00. 

Ans.  $  53.79. 

27.  How.  many  cubic  inches  are  contained  in  a  cube  that  may 
be  inscribed  in  a  sphere  40  inches  in  diameter  ? 

Ans.  12316.8+  inches. 

28.  A  bushel  measure  is  18i  inches  in  diameter,  and  8  inches 
deep ;  what  should  be  the  dimensions  of  a  similar  measure  that 
would  contain  4  quarts  ? 

Ans.  9|-  inches  in  diameter,  4  inches  deep. 

29.  A  gentleman  willed  -J  of  his  estate  to  his  wife,  and  |  of 
the  remainder  to  his  oldest  son,  and  ^  of  the  residue,  which  was 
$  151.33^,  to  his  oldest  daughter;  how  much  of  his  estate  is  left 
to  ber*divided  among  his  other  heirs?  Ans.  $756.66f. 

30.  A  man  borjueathed  ^  of  his  estate  to  his  son,  and  ^  of  the 
remainder  to  his  daughter,  and  the  residue  to  his  wife ;  the  differ- 
ence between  his  son  and  daughter's  portion  was  $  100 ;  what 
did  he  give  his  wife  ?  Ans.  $  600.00. 

31.  Sold  a  lot  of  shingles  for  $50,  and  by  so  doing  I  gained 
12^  per  cent.  ;  what  was  their  value  ?  Ans.  $  44.44|. 

32.  If  -^j  of  a  yard  cost  $  5.00,  what  quantity  will  $  17.50  pur- 
chase ?  Ans.  IJ-  yard. 

33.  John  Savory  and  Thomas  Hardy  traded  in  coni)*(Uiy ; 
Savory  put  in  for  capital  $  1000  ;  they  gained  $  128.00  ;  Hardy 
received  for  his  share  of  the  gains  $  70  ;  what  was  his  capital  ? 

Ans.  %  1206.80^1. 


322  MISCELLANEOUS    QUESTIONS. 

34.  E.  Fuller  lent  a  certain  sum  of  money  to  C.  Lamson,  and 
at  the  end  of  3  year-s,  7  months,  and  20  days  he  received  mterest 
and  principal  $  1000  ;  what  was  the  sum  lent? 

An.?.  $820.79f|i. 

35.  Lent  $88  for  18  months,  and  received  for  interest  and 
principal  $  97.57  ;  what  was  the  per  cent.  ?     Ans.  7^  per  cent. 

36.  When  f  of  a  gallon  cost  S  87,  what  cost  7^  gallon-^  ? 

Ans.  $  1051.25. 
■37.  When  $  71  are  paid  for  18f  yards  of  bi-oadcloth,  what  cost 
5  yards  ?  Ans.  $  19.26-^^^ 

38.  How  many  yards  of  cloth,  at  $  4.00  per  yard,  must  be  given 
for  18  tens  17cwt.  3qr.  of  sugar,  at  S  9.50  per  cwt.  ? 

Ans.  897^^2-  yards. 

39.  How  much  grain,  at  $  1.25  per  bushel,  must  be  given  for 
98  bushels  of  salt,  at  $  0.45  per  bushel  ?       Ans.  35/^  bushels. 

40.  A  per.-on  being  a>ked  the  time  of  day,  replied  tliat  |  of  the 
time  passed  from  noon  was  equal  to  j\  of  the  time  to  midnight. 
Required  the  time.  Ans.  40  minutes  past  4. 

41.  On  a  certain  night,  in  the  year  1852,  rain  fell  to  the  depth 
of  3  inches  in  the  town  of  Haverhill  :  the  town  contains  about 
20,000  square  acres,  llequired  the  number  of  hogsheads  of  water 
fallen,  supposing  each  hogshead  to  contain  100  gallons,  and  each 
gallon  282  cubic  inches. 

Ans.  1334G042hhd.  55gah  Iqt.  Opt.  2^flgi. 

42.  If  the  .sun  pass  over  one  degree  in  4  minutes,  and  the  lon- 
gitude of  Boston  is  71°  4'  west,  what  Avill  be  the  time  at  Boston, 
when  it  is  llh.  IGm.  A.  M.  at  London  ? 

Ans.  Gh.  31m.  44sec.  A.  M. 

43.  When  it  is  2h.  3Gm.  A.  M.  at  the  Cape  of  Good  Hope,  in 
longitude  18°  24'  east,  what  is  the  time  at  Cape  Horn,  in  longi- 
tude G7°  21'  west?  Ans.  8h.  53m.  P.  M. 

44.  Yesterday  my  longitude,  at  noon,  was  1G°  18'  west ;  to-day 
I  perceive  by  my  Avatcli,  which  has  kept  correct  time,  that  tlie 
sun  is  on  the  meridian  at  llh.  36m. ;  what  is  my  longitude  to- 
day ?  Ans.  10°  18' west. 

45.  Sound,  imintorruptcd,  will  pa^s  1142  feet  in  1  second  ;  how 
long  will  it  be  in  i)a-sing  from  Boston  to  London,  the  distance 
being  al)out  3000  miles  ?  Ans.  3h.  51m.  10-f-  sec. 

46.  The  time  whic-h  elap.sed  between  seeing  the  flash  of  a  gun 
and  tiearing  its  report  was  10  seconds;  wbat  was  the  distance  ? 

Ans.  2  miles  860  feet. 

47.  J.  Pearson  has  tea,  which  he  barters  with  M.  Swift,  at 


MiscEi  ;aneous  quei^tions.  323 

10  cents  per  lb.  more  than  it  costs  him  for  sugar,  which  costs 
Swift  15  cents  per  pound,  but  which  he  puts  at  20  cents  per 
pound ;  what  was  the  first  cost  of  tlie  tea  ? 

Ans.  $  0.30  per  lb. 

48.  Q  and  Y  barter;  Q  makes  of  10  cents  12^  cents;  Y 
makes  of  15  cents  19  cents ;  which  makes  the  most  per  cent.,  and 
how  much?  Ans.  Y  makes  1§  per  cent,  more  than  Q. 

49.  A  certain  individual  was  born  in  178G,  September  25,  at 
23  minutes  past  3  o'clock,  AM.;  how  many  minutes  old  will 
he  be  July  4,  1844,  at  30  minutes  past  5  o'clock,  P.  M.,  reck- 
oning 365  days  for  a  yeai',  excepting  leap  years,  wiiich  have  366 
days  each  ?  Ans.  3038(^287  minutes. 

50.  The  longitude  of  a  certain  star  is  33.  14°  26'  14",  and  the 
longitude  of  the  moon' at  the  same  time  is  8s.  19°  43'  28";  how 
far  will  the  moon  have  to  move  in  her  orbit  to  be  in  conjunction 
with  the  star  ?  Ans.  6s.  24°  42'  46". 

51.  From  a  small  field,  containing  3A.  IR.  23p.  200ft.,  there 
were  sold  lA.  2R.  37p.  30yd.  8ft. ;  what  quantity  remained  ? 

Ans.  lA.  2R.  25p.  21yd.  5ft.  36in. 

52.  What  part  of  |  of  an  acre  is  |  of  an  acre  ? 

Ans.  f^. 

53.  A  thief  was  brought  before  a  certain  judge,  and  it  was 
proved  that  he  had  stolen  property  to  the  value  of  l£  19s.  llfd. 
He  was  sentenced  either  to  one  year's  imprisonment  in  the  county 
jail,  or  to  pay  l£  19s.  lljd.  for  the  value  of  every  pound  he 
had  stolen  ;  required  the  amouijl  of  the  fine. 

Ans.  3£  19s.  lid.  Og-^^jqr. 

54.  My  chaise  having  been  injured  by  a  very  bad  boy,  I  am 
obliged  to  sell  it  for  $  68.75,  which  is  40  per  cent,  less  than  its 
original  value  ;  what  was  the  cost?  Ans.  S  114.58^. 

55.  Charles  "Webster's  horse  is  valued  at  $  120,  but  he  will  not 
sell  him  for  less  than  $  134.40  ;  what  per  cent,  does  he  intend  to 
make?  Ans.  12  per  cent. 

56.  Three  merchants,  L.  Emerson,  E.  Bailey,  and  S.  Curtiss, 
engage  in  a  cotton"  speculation.  Emerson  advanced  $3600, 
Bailey  $4200,  and  Curtiss  $2200.  They  invested  their  whole 
capital  in  cotton,  for  which  they  received  $  15000  in  bills  on  a 
bank  in  New  Orleans.  These  bills  were  sold  to  a  Boston  broker 
at  15  per  cent,  below  par  ;  what  is  each  man's  net  gain  ? 

Ans.  Emerson  $  990.00,  Bailey  $  1155.00,  Curtiss  $  605.00. 

57.  Bought  a  box  made  of  plank,  3J  inches  thick.  Its  length 
on   the   outside   is   4ft.  9in.,  its   bvcadai   3ft.  7in.,'  and  its  Iiight 


824  MiscEii,,AXEous  queIttions. 

2ft.  11  in.     How  many   square  feet  did  it  require  to  make  the 
box,  and  how  many  cubic  feet  will  it  hold  ? 

Ans.  10^^  square  feet,  29^  cubic  feet. 

58.  How  many  bricks  will  it  require  to  construct  the  walls  of 
a  house  64  feet  long,  and  32  feet  wide,  and  28  feet  high  ?  The 
walls  are  to  be  1ft.  4in.  thick,  and  there  are  also  three  doors  7ft. 
4in.  high,  and  3ft.  8in.  wide  ;  also  14  windows  3  feet  wide  and  6 
feet  high,  and  16  windows  2ft.  8in.  wide  and  5ft.  8in.  high.  Each 
brick  is  to  be  eight  inches  long,  and  4  inches  wide,  and  2  inches 
thick.  Ans.  167,480  bricks. 

59.  John  Brown  gave  to  his  three  sons,  Benjamin,  Samuel, 
and  William,  $  1000,  to  be  divided  in  the  pi'oportion  of  ^,  ^,  and 
■J,  respectively  ;  but  William,  having  received  a  fortune  by  his 
w^ife,  resigns  his  share  to  his  brothers.  It  is, required  to  divide  the 
whole  sum  between  Benjamin  and  Samuel. 

Ans.  Benjamin,  $571.42f ;  Samuel,  $428.57f. 

60.  Peter  Webster  rented  a  house  for  1  year  to  Thomas  Bai- 
ley, for  $  100  ;  at  the  end  of  four  months  Bailey  rented  one  half 
of  the  house  to  John  Bricket,  and  at  the  end  of  eiglit  months  it 
was  agreed  by  Bricket  and  Bailey  to  rent  one  third  of  the  house 
to  John  Dana.     What  share  of  the  rent  must  each  pay  ? 

Ans.  Bailey  $  61^,  Bi'icket  $  27^,  and  Dana  $  11^. 

61.  I  have  a  plank  42|-  feet  in  length,  24  inches  wide,  and  3 
inches  thick  ;  required  the  side  of  a  cubical  box  that  can  be  made 
from  it.  Ans.  48  inches. 

62.  D.  Small  purchased  a  horse  for  10  per  cent,  less  than  his 
value,  and  sold  him  for  1 6  per  cent,  more  than  his  value,  by  which 
he  gained  $  21.84 ;  what  did  he  pay  for  the  horse  ? 

63.  Minot  Thayer  sold  broadcloth  at  $  4.40  per  yard,  and  by 
so  doing  he  lost  12  per  cent. ;  whereas,  he  ought  to  have  gained 
10  per  cent.  ;  for  what  should  the  cloth  have  been  sold  per  yard? 

64.  A  gentleman  has  5  daughters,  Emily,  Jane,  Betsey,  Abi- 
gail, and  Nancy,  whose  fortunes  are  as  follows.  The  first  two 
.and  the  last  two  have  $19,000;  the  first  four,  $  19,200;  the 
last  four,  $  20,000  ;  the  first  and  the  last  three,  $  20,500 ;  the  first 
three  and  the  last,  $21,300.     What  was  the  fortune  of  each? 

Ans.  Emily  has  $5,000;  Jane,  $4,500;  Betsey,  $6,000, 
Abigail,  $  3,700  ;  and  Nancy,  $  5,800. 

65.  I  have  a  fenced  garden,  12  rods  square.  How  many  trees 
may  be  set  on  it,  whose  distance  from  each  other  shall  be  one  rod, 
and  no  tree  to  be  within  half  a  rod  of  the  fence  ?  A^ns.  152  trees. 


•^'.'.' .' 


>        r 


•"  * 


>^ 


^^^^ 


/    ; 


/  -  ' 


GRTTfiJNLEAF'S 

MATKFU4      OIL    SEEIES, 

ADAPTLb    J  >         '     CLASSES   OF    LEARNERS. 


For  District  S,l..oc  Ji*. 

NEW  PRIMi^RY  ARiTUi.:T[C. 
NEW  IXTELLEOTIJAI  mTiIMETIC. 
COJDION  SCHOOL  AKITDMETrC. 


High  Schools  and  Academies. 

NATIONAL  ARITHMETIC. 
SEW  ELEMENTARY  ALGEBRA. 
ELEMENTS  OF  GEOMETRY. 


For  Academies,  Normal  Schools,  ar.d  CoUeg-es. 

NEW   fllGHER   ALGEBRA,  ELEMENTS  OF  TRIGONOMETRY. 

ELEMENTS  OF  GEOMETRY  AND  TRIGONOMETRY. 


The  NEW  PRIilARY  AIUTHIIETIC  is  an  attractive  and  interesting 
book  of  easy  lessons  for  beginner^,  adapted  to  the  Object  Method. 

The  NEW  INTELLECTUAL  ARITHMETIC  is  a  late  work,  wifli 
MODEiwS  of  ANALYSIS,  for  Common  Schools  and  Academies. 

The  COMMON  SCHOOL  ARITH.METIC ;  or  a  complete  system 
of  Written  Arithmetic  for  Common  Schools  and  Seminaries,  is  a  prac- 
tical and  common-sense  treatise,  being  sufficient  to  prepare  the  learner 
f<i>  all  ordinary  business.     .  ,, 

O^The  NATIONAL   ARITHMETIC  h  a  thorough  advanced  course  for 
>itgh  Scliools,  Academies,  Normal  Schools,  and  Commercial  Colleges. 

The  NEW  ELEMENTARY  ALGEBRA  contains  the  iirst  princi- 
ples of  Analysis  progressively  developed  and  simplified,  for  Common 
Schools  and   Academies. 

The  NEW  HIGHER  ALGEBRA  is  a  superior  course,  on  the  plan 
of  the  last-named  work,  but  especially  adapted  to  more  advanced  classes. 

The  ELEMENTS  OF  GEOMETRY,  with  Applications  to  Mensura- 
tion, and  many  practical  exercises. 

The  ELEMENTS  OF  TRIGONOMETRY,  with  Applications,  is  an 
elegant  treatise,  in  accordance  with  the  modern  Ratio  Method. 

The  GEOMETRY  AND  TRIGONOMETRY,  consists  of  the  'iust- 
uamed  two  works,  included  in  one  volume. 

q;;^-  All  the  above-named  books  are  new,  except  tlia  Coii.n.on 
School  and  National  Arithmoticffj  which  are  standard  works,  rf"-;'  -' 

[X^  The  fact  that  GnERNLEAF's  Mathematical  Sehies  ■  is  .uw 
u'ed  In  most  of  the  St.vtk  Nohmal  Schools,  as  well  as  in  xii  the 
]>'■  •    -Ks  ill  the  United  States,  is  the  highest  recommendation. 

z  1  by  ROBERT  S.  DAVIS  &  CO.,  Boston. 


I 


